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1. 5x y 9x3y6 8x y 36a b 48a b 64a p T OO 9ab 1206 16b 4a b PROBLEM SET 0 3 In Problems 1 10 perform the indicated operations 9 5 x 2 4 x 3 2 x 6 1 5x 7x 2 9x 8x 4 10 3 2x 1 2 3x 4 4 5x 1 23 2 o c a E In Problems 11 54 find the indicated products Remem 3 14x x 1 15x 3x 8 ber the special patterns that we discussed in this section 4 3x 2x 4 4x 6x 5 11 3xy 4Ax y 5xy 5 3x 4 6x 3 9x 4 12 2ab 3a b 4ab 6 7a 2 8a 1 10a 2 13 6a b Sab 4a b 3ab 7 8x 6x 2 x x 1 3x 2x 4 14 xy 5x y 4xy 3x y 8 12x 7x 2 3x Ax 5 4x 7x 2 15 x 8 x 12 16 x 9 x 6 Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 36 17 19 21 23 25 27 29 30 31 32 33 34 35 36 37 38 39 40 41 43 45 47 49 51 53 For Problems 55 66 use Pascal s triangle to help ex Chapter O Some Basic Concepts of Algebra A Review n 4 n 12 s t x y 3x 1 2x 3 4x 3 3x 7 x 4 Qn 3 x 2 x 4 x 3 x 1 x 6 x 5 18 20 22 24 26 28 x
2. 61 4 V2 V3 3V2 2V3 62 2V6 3V S3V 6 4V5 63 6 2V5 6 2V5 64 7 3V2 7 3V2 65 Vx Vy 66 2Vx 3V yy 67 Va Vb Va Vb 68 3 Vx 5Vy GVx SVy For Problems 69 80 rationalize the denominator and simplify All variables represent positive real numbers 3 7 69 MN NE he MS 42 4 10 3 4 2 di n v V3 V5 V3 we Ael PIE AN 2V5 37 5V2 33 5 as _V _ lt Vx 1 Vx 2 Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 70 TT 79 Chapter O Some Basic Concepts of Algebra A Review _ Vx Vy 2Vx Vy 3Vx 2Vy 2Vx USE 3Vx 2V y 2Vx SV y For Problems 81 84 rationalize the numerator All vari ables represent positive real numbers 81 V 2x 2h NV 2x h H5 Thoughts into words 85 86 87 Is the equation Vx y x V y true for all real number values for x and y Defend your answer Is the equation V x y xy true for all real number values for x and y Defend your answer Give a step by step description of how you would change V 252 to simplest radical form 82 83 84 88 89 Vx t h 4 Vx 1 Vath h 2Vx h 2Vx h Vx 3 Why is V 9 not a real number How could you find a whole number approximation for V 2750 if you di
3. 12a 1 42 18n 39n 15n 8x 2xy y 44 12x 7xy 10y 2n n 5 46 25t 100 2n 14n 20n 48 25n 64 Ax 32 50 2x 54 x 4x 45 52 x x 12 2x y 26x y 96y 54 3x y 15x y 108y a by c dy a b c d x 8x 16 y 58 4x 12x 9 y x y 10y 25 60 y x 16x 64 60x 32x 15 62 40x 37x 63 84x 57x 60x 64 210x 102x 180x For Problems 65 74 factor each of the following and assume that all variables appearing as exponents repre sent integers 65 67 69 71 73 x 16 66 x 9 xan ym 68 x y x 3x 28 70 x 10x 21 2x Tx 30 72 3x 16x 12 74 16x 241 9 H5 Thoughts into words 76 77 78 79 Describe in words the pattern for factoring the sum of two cubes What does it mean to say that the polynomial x 5x 7 is not factorable using integers What role does the distributive property play in the factoring of polynomials Explain your thought process when factoring 30x 13x 56 75 80 0 4 Factoring Polynomials 47 Suppose that we want to factor x 34x 288 We need to complete the following with two numbers whose sum is 34 and whose product is 288 x 34x 288 x __ x _ These numbers can be found as follows Because we need a product of 288 let s consider the prime
4. 51 ab c 52 abet 53 2x y y 54 3x y 2 2 4 3 55 56 25 y x 55744 3 3x2y 57 58 im 4a b y a 59 60 25 ab Ja olin 61 Ei 62 PORE ve For Problems 63 70 find the indicated products quo tients and powers express answers without using zero or negative integers as exponents 63 4x y 5xy 64 6xy 3x y 65 3xy y 66 2x y 4 2 2 3 4 3 67 68 35 y y 72x 108x 69 P 9x 12x 70 For Problems 71 80 find the indicated products and quotients express results using positive integral expo nents only 71 2x 1y 3x y 73 6 y a 7y 24x ly a x 72 Ax 7y3 5x3y 74 74 8a b 60 1b 56xy gt 8x7 76 Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 35gb 27a b T 78 7a b 3a p 14x 2y V 2 24x5y 2 N73 79 jy 80 Er y x y For Problems 81 88 express each as a single fraction in volving positive exponents only 81 x 82 x 84 2x7 3y73 85 3a 2b 2 86 a a gt 1 2 87 x y 307 88 x y x y For Problems 89 98 find the following products and quotients Assume that all variables appearing as expo nents represent integers For example Cx Om 2b tO b x5 89 3x 4x 4 90 5x 6x
5. y 2x i 5y 6x Solution B The LCD of all four denominators x y x and y is xy Let s multiply the entire complex fraction by a form of 1 namely xy xy ON N a 1M amp 2 lt N a1 amp w o3 w a S 3y 2xy y 3y 2x ar Le UBI H Sy 6x 5y 6x Certainly either approach Solution A or Solution B will work with a prob lem such as Example 7 We suggest that you study Solution B very carefully This approach works effectively with complex fractions when the LCD of all the Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 56 Chapter O Some Basic Concepts of Algebra A Review denominators is easy to find Let s look at a type of complex fraction used in cer tain calculus problems td Solution SD 11 pod x h x 72 x h x h x x h h 1 1 1 1 E x x 4 n x x 4 x x h h x x h x x h hx x h hx x h h 1 x x h x x h Example 9 illustrates another way to simplify complex fractions 1 a n Solution We first simplify the complex fraction by multiplying by n n n 2 m m n n 1 n 1 Now we can perform the subtraction 1 n G n n 1 n 1 1 n 1 2 72 1 n 72 1 n i n 1 m mtn 1 a or n 1 n 1 Finally we need to recognize that complex fractions are so
6. 3x 7x 1 You can also think in terms of the property x 1 x and the distributive prop erty Therefore 3x 7x 1 1 3x 7x 1 3x 7x 1 Now consider the following subtraction problems 7x 2x 4 3x 7x 1 Ux 2x 4 3x 7x 1 7x 3x 2x 7x 4 1 4x 9y 3 Ars arm y ey Ay 2 yo y 2 om B D 7 2 Ty y 9 Multiplying Polynomials The distributive property is usually stated as a b c ab ac but it can be ex tended as follows a b c d ab ac ad a b ct d t e ab ac ad ae etc The commutative and associative properties the properties of exponents and the distributive property work together to form the basis for finding the prod uct of a monomial and a polynomial with more than one term The following ex ample illustrates this idea 3x 2x 5x 3 3x7 2x 3x 5x 3x 3 6x4 15x 9x Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 30 Chapter O Some Basic Concepts of Algebra A Review Extending the method of finding the product of a monomial and a polyno mial to finding the product of two polynomials each of which has more than one term is again based on the distributive property x 2 y 5 2 x y 5 2 y 5 x y x 5 2 y X
7. 4 500 Qi _4i 5i 2 _ 4i S 1 2 1 PROBLEM SET 0 8 For Problems 1 14 add or subtract as indicated 1 5 27 8 60 2 9 3i 4 amp 5i 3 8 670 5 27 4 6 4i 4 6i 5 7 37 4 4i 6 6 7i 7 6i 7 2 3i 1 i 8 3 eG ii i i 5 1 7 1 Ezi ti H 5 5i G zi 3 3 NER 11 3 5 T s 12 4 i V3 6 2i V3 13 5 3i 7 2i 8 i 14 5 7i 6 2i 1 2i For Problems 15 30 write each in terms of i and sim plify For example V 20 iV20 iV4N 5 215 15 V 9 16 V 49 17 V 19 18 V 31 19 ES 20 x 21 V 8 22 V 18 23 V 27 24 V 32 25 V 54 26 V 40 27 3V 36 28 5V 64 29 4V 18 30 6V 8 For Problems 31 44 write each in terms of i perform the indicated operations and simplify For example vV 9V 16 iV9 iV16 3i 4i 12 12 1 12 31 V 4V 16 32 V 25V 9 33 V 2V 3 34 V 3V 7 Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 35 VZ5V 4 36 V 7V 9 37 V 6V 10 38 V 2V 12 39 V 8V 7 40 V 12V 5 ai 6 9 d 16 54 18 43 44 4 0 V 3 For Problems 45 64 find each product and express the answe
8. 48 onm 30x7 6 8 6 y 17 16x 13 206 10 8 11 12x y 277 4y 1 e x ab y y 3x 8 21 12a 19 18 19 4x 1 3 22 20x 11x 42 12x 17x 6 xX x 19x 28 24 35x 22x 3 26 6x x 10x 6 25x 30x 9 28 9x 42x 49 8x 12x 6x 1 27x 135x 225x 125 x 2x3 6x 22x 15 2x 11x 1632 8x 8 33 4x2 y 8xy 7y Oxy 35 3x 2y 3x 2y Bx 19 EESZROX 5 L Ge yeas Ed vie 3 39 x 2 x y 4x 3y 16x 12xy 9y 83x 45x 2 42 3 x3 12 Notfactorable 44 3 x 2 x 2x 4 x 3 x 3 Y x 2 x 2 2x 1 y 2x 14 5a 3 29x 10 12 3x 16 29 30 31 axe 3y 2 3x 1 50 x 4 x 38 6n 15 53 15 5n 3x 8x 40 y 3x 3S x 48 49 51 52 54 55 56 61 64 x x 7 8x 4 x x 2 x 2 3x 2 4x 3 20V3 V3 65 24 4V6 15 i x hy 62 6x V 6x V 10x 2y 2 3x 6V xy 68 x 4 x 4 x 10 3xy 2x 7 Sys TX 6x 3h 12 0 x 2y 63 2xy V dzy 15 3V2 6 s 3 69 V 55 x 41 Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated
9. 1 2x 3 3x 2 2x 5 x 4 3x 1 x 1 x 3x 4 t D 2t 4 t DP 9 t1 2x 1 x 4x 3 3x 2 2x x 1 3x 2 2x 3x 4 n 6 n 10 a b ct d 5x 2 3x 4 4n 3 6n 1 x 6 Bn 5 2 x 2x 1 x 6x 4 x x 4 2x 3x 5x 2 5x 2 x 5x 2 2x 3y Qx 3y x 5 2x 1 4x 3 5x 20 pand each expression 55 56 a by a b 44 46 48 50 52 54 1 42 3x 4 3x 4 x 4 1 9x y 9x y per 3x 4y 2x 5y x 3yy 57 x yy 58 x yy 59 x 217 60 2x yp 61 2a 0 62 3a 0 63 x yy 64 x 2y y 65 2a 3b 66 4a 3b In Problems 67 72 perform the indicated divisions 6 15x 25x 68 48x5 72x 5x i 8x 300 24d 540 18x3y 27x2y 6a 3xy 20a b 44a b gt 4a b 71 2133y8 28x4y gt 355 y 72 dy y In Problems 73 82 find the indicated products As sume all variables that appear as exponents represent integers 73 x yP x y 74 x2 1 x 3 75 x 4 x 7 76 3x 2 x 5 77 2x 1 8x 2 78 Qx 3 2x 3 79 x 1 80 x 2 81 x 2 82 x 3p Copyright 2005 Thomson Learning Inc All Rights Reserved May
10. 1 has a hole at o 5 because x cannot equal zero When you use a graphing cal culator this hole may not be detected Except for the hole the graphs are identi 8 2 2 cal and we are claiming that 77 f for all values of x except 0 and 4 x 4x x 4 10 10 10 10 Figure 0 20 io wt ont 5 722 777 8 Solution f 672 5 02 5 07 1 m In 8 n 8 07 1 j LCD n 1 n 5 n 8 mos ae a e e 3n n 8 n SY n 1 n 8 4 n 5 n 5 n 1 n 8 3n 24n 4n 20 n SY n 1 n 8 _ 3n 2077 20 n 5 n 1 n 8 Simplifying Complex Fractions Fractional forms that contain rational expressions in the numerator and or the de nominator are called complex fractions The following examples illustrate some approaches to simplifying complex fractions Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 0 5 Rational Expressions 55 3 2 2o eg KO Simplity x y Solution A Treating the numerator as the sum of two rational expressions and the denomina tor as the difference of two rational expressions we can proceed as follows M s F QUNM Ay E 3 OG GE x y XI Ny YAK 3y 2x 3y 2x _ Xy xy xy 5y 6x 5y 6x xy xy xy y 3y 2x xy 5y 6x
11. 8 7b 9a 6b a 7 and b 8 5x 8y Ty 8x x 5 and y 6 x y x 4 and y 7 3x y 2x 4y x 5 and y 3 x y y x x 6and y 13 Thoughts into words 80 81 82 Do you think 3V2 is a rational or an irrational num ber Defend your answer 0 8 Explain why ri 0 but 0 is undefined The solution of the following simplification prob lem is incorrect The answer should be 11 Find and correct the error 76 TT 78 79 83 0 2 Exponents 17 a 4andb 8 5S x 1 7 x 4 x 3 2 3x 4 3 2x 1 x 2 4 2x 1 5 3x T x 1 5 a 3 4 2a 1 2 a 4 a 3 8 4 2 3 4 2 1 2 2 12 1 4 12 16 Explain the difference between simplifying a nu merical expression and evaluating an algebraic expression E Graphing calculator activities 84 Different graphing calculators use different se quences of key strokes to evaluate algebraic expres 0 2 Exponents sions Be sure that you can do Problems 59 79 with your calculator Positive integers are used as exponents to indicate repeated multiplication For ex ample 4 4 4 can be written 4 where the raised 3 indicates that 4 is to be used as a factor three times The following general definition is helpful DEFINITION 0 2 If n is a positive integer and b is any real numbe
12. Ib palermo Ti T3 Ga Figure 0 10 Figure 0 11 Historically the rectangular coordinate system provided the basis for the de velopment of the branch of mathematics called analytic geometry or what we presently refer to as coordinate geometry In this discipline Ren Descartes a French 17th century mathematician was able to transform geometric problems into an algebraic setting and then use the tools of algebra to solve the problems Basically there are two kinds of problems to solve in coordinate geometry 1 Given an algebraic equation find its geometric graph 2 Given a set of conditions pertaining to a geometric figure find its algebraic equation Throughout this text we will consider a wide variety of situations dealing with both kinds of problems For most purposes in coordinate geometry it is customary to label the hori zontal axis the x axis and the vertical axis the y axis Then ordered pairs of real numbers associated with points in the xy plane are of the form x y that is x is the first coordinate and y is the second coordinate Properties of Real Numbers As you work with the set of real numbers the basic operations and the relations of equality and inequality the following properties will guide your study Be sure Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 10 Chapter O Some Basic Concep
13. __ 3x _ Because the factors of the last term 2 are 1 and 2 we have only the following two possibilities to try x 2 3x 1 or x 1 3x 2 By checking the middle term formed in each of these products we find that the sec ond possibility yields the desired middle term of 5x Therefore 3x 5x 2 x 1 3x 2 a EXAMPLE 5 Factor 8x 30x 7 Solution First observe that the first term 8x can be written as 2x 4x or x 8x Second be cause the middle term is negative and the last term is positive we know that the bi nomials are of the form 2x __ 4x or oaae Third because the factors of the last term 7 are 1 and 7 the following possibilities exist 2x 1 4x 7 2x 7 4x 1 x 1 8x 7 x 7 8x 1 By checking the middle term formed in each of these products we find that 2x 7 4x 1 produces the desired middle term of 30x Therefore 8x 30x 7 2x 7 4x 1 B EXAMPLE 6 Factor 5x 18x 8 Solution Ds p The first term 5x can be written as x 5x The last term 8 can be written as 2 4 2 4 71 8 or 1 8 Therefore we have the following possibili ties to try x 2 5x 4 x 4 5x 2 x 1 5x 8 x 8 5x 1 x 2 5x 4 x 4 5x 2 x 1 5x 8 x 8 5x 1 Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scann
14. 1N2 36 1p4 1 17 E2 18 35 2x y 12a b For Problems 19 34 perform the indicated operations 19 7x 3 Sx 2 6x 4 20 12x 5 7x 4 8x 1 21 3 a 2 2 3a 5 3 5a 1 22 4x 7 5x 6 23 3x 2 4x 3 24 Tx 3 5x 1 25 x 4 x 3x 7 26 2x 1 3x 2x 6 27 5x 3 28 3x 7 29 2x 1 30 3x 5 31 x 2x 3 x 4x 5 32 2x x 2 x 6x 4 24 y 48x y 56x y 7223 34 zu 6xy 8x 33 May not be copied scanned or duplicated in whole or in part Licensed to iChapters User Chapter O Review Problem Set 87 For Problems 35 46 factor each polynomial completely 60 Simplify the expression Indicate any that are not factorable using integers d ibn 6 x 2 6ext x 2 1 35 9x 4y 36 3x 9x 120x x 3 aye 37 4x 20x 25 38 x y 9 39 x 2x xy 2y 40 64x3 2713 For Problems 61 68 express each in simplest radical form All variables represent positive real numbers 41 15x 14x 8 42 3x 36 61 5V 48 62 3 V24 43 2x x 8 44 3x 24 45 x 13x 36 46 A 4x 1 y 13321 3V8 d y 63 V32x y 64 2 V6 For Problems 47 56 perform the indicated operations 5x 3 65 J 66 involving rational expressions E
15. 21 d 17 and 42 e 56 and 21 f 0 and 37 44 Evaluate each of the following if x is a nonzero real number x x Ll b x x d x x In Problems 45 58 state the property that justifies each of the statements For example 3 4 4 3 because of the commutative property of addition 45 x 2 2 x 46 7 4 6 7 4 6 47 l x x 48 43 18 18 43 49 1 93 93 50 109 C109 0 51 5 4 7 5 4 5 7 52 l x y x y 53 7yx 73 54 x 2 22 x 2 2 55 6 4 7 4 6 7 4 2 3 s 5 2 57 4 5x 4 5 x 58 07 8 5 17 8 25 For Problems 59 79 evaluate each of the algebraic ex pressions for the given values of the variables 59 5x 3y x 2 and y 4 60 7x 4y x 1andy 6 61 3ab 2c a 4 b 7 and c 8 62 x 2y 3z x 3 y 4 and z 9 63 a 2b 3c 4 a 6 b 5 and c 11 64 3a 2b Ac 1 a 4 b 6 and c 8 2x F Ty 5 oes aad c ud x y diy 12 UC Wee TAT 67 5x 2y 3x 4y x 3andy 7 68 2a 7b 4a 3b a 6and b 3 Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 5x 5a x yl 4y 9y 2y x 2andy
16. The number 7 is not a real number and is often called the imaginary unit but the number i is the real number 1 The imaginary unit i is used to define a complex number as follows DEFINITION O 8 A complex number is any number that can be expressed in the form QD where a and b are real numbers and i is the imaginary unit The form a bi is called the standard form of a complex number The real number a is called the real part of the complex number and b is called the imagi nary part Note that b is a real number even though it is called the imaginary part Each of the following represents a complex number 6 t 2i is already expressed in the form a bi Traditionally com plex numbers for which a 0 and b 0 have been called imaginary numbers yeg can be written 5 3 even though the form 5 3i is often used 8 iv2 can be written 8 V 2i It is easy to mistake V 2i for V 2i Thus we commonly write i V2 instead of V2i to avoid any difficulties with the radical sign 9i can be written 0 9i Complex numbers such as 9i for which a 0 and b 0 traditionally have been called pure imaginary numbers 5 can be written 5 07 The set of real numbers is a subset of the set of complex numbers The fol lowing diagram indicates the organizational format of the complex number system Complex numbers a bi where a and b are real numbers ond ia Real numbers Imaginary numbers a
17. in whole or in part 0 472195631 0 21411711191111 Nonrepeating decimals 0 752389433215333 A repeating decimal has a block of digits that repeats indefinitely This repeating block of digits may be of any size and may or may not begin immediately after the decimal point A small horizontal bar is commonly used to indicate the repeating block Thus 0 3333 can be expressed as 0 3 and 0 24171717 as 0 2417 In terms of decimals a rational number is defined as a number with either a terminating or a repeating decimal representation The following examples illus a trate some rational numbers written in b form and in the equivalent decimal form 3 3 1 1 1 E rix 0 75 1 0 27 ET 0 125 37 0 142857 a 0 3 er a We define an irrational number as a number that cannot be expressed in b form where a and b are integers and b is not zero Furthermore an irrational number has a nonrepeating nonterminating decimal representation Following are some examples of irrational numbers They are decimals that do not terminate and do not repeat V2 1 414213562373095 V3 1 73205080756887 m 3 14159265358979 The entire set of real numbers is composed of the rational numbers along with the irrationals The following tree diagram can be used to summarize the var ious classifications of the real number system Real numbers Rational Irrational Integers Nonintegers S 0 amp a Any real nu
18. then bin Wb Definition 0 6 states that b means the nth root of b We shall assume that b and n are chosen so that Wb exists in the real number system For example 25 is not meaningful at this time because V 25 is not a real number The fol lowing examples illustrate the use of Definition 0 6 251 2 V25 5 16 V16 2 813 Y8 2 27 5 W 27 3 Now the following definition provides the basis for the use of a rational numbers as exponents Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 72 Chapter O Some Basic Concepts of Algebra A Review DEFINITION 0 7 If m n is a rational number expressed in lowest terms where n is a positive in teger greater than 1 and m is any integer and if b is a real number such that Wb exists then pin Np N p In Definition 0 7 whether we use the form Wb or V b for computational pur poses depends somewhat on the magnitude of the problem Let s use both forms on the following two problems 823 Y8 64 4 or 8 178 2 4 2733 V27 729 9 o 2DT5 Yoy To compute 8 both forms work equally well However to compute 277 the form v 27 is much easier to handle The following examples further illustrate Definition 0 7 2532 25 5 125 1 1 1 1 32 35 25 Nap 2 4 64y N 64y 4 1
19. y x 3 3 9 x 3 7 3 y 4x 2y 1 2x 2y 1 2x 2y 1 2x 2y 1 2x 2y 1 x 1 x 4 i 1 x 4x D x 4 x 1 x 4 x 1 x 4 2x 3 5 It is possible that both the technique of factoring out a common monomial factor and the pattern of the difference of two squares can be applied to the same problem Jn general it is best to look first for a common monomial factor Consider the following examples 2x 50 2 x 25 2 x 5 x 5 48y 27y 3y 16y 9 3y 4y 3 4y 3 9x 36 9 x 4 9 x 2 x 2 Factoring Trinomials Expressing a trinomial as the product of two binomials is one of the most common factoring techniques used in algebra As before to develop a factoring technique we Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User EXAMPLE 1 ys EXAMPLE 3 0 4 Factoring Polynomials 41 first look at some multiplication ideas Let s consider the product x a x b using the distributive property to show how each term of the resulting trinomial is formed x t a x t b 2 x x b a x b x x x b a x a b x a b x ab Notice that the coefficient of the middle term is the sum of a and b and that the last term is the product of a and b These two relationships can
20. 0 3 we called your attention to some special multiplication patterns One of these patterns was a b a b a b This same pattern viewed as a factoring pattern Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 40 Chapter O Some Basic Concepts of Algebra A Review is referred to as the difference of two squares Applying the pattern is a fairly simple process as these next examples illustrate Again the steps we have included in dashed boxes are usually performed mentally x 16 xF 4 x 4 x 4 4x 25 x 5 2x 5 2x 5 Because multiplication is commutative the order in which we write the factors is not important For example x 4 x 4 can also be written x 4 x 4 You must be careful not to assume an analogous factoring pattern for the sum of two squares it does not exist For example x 4 x 2 x 2 because x 2 x 2 2 x Ax 4 We say that a polynomial such as x 4 is not factorable using integers Sometimes the difference of two squares pattern can be applied more than once as the next example illustrates 16x 81y 4x 97 Ax 9y Ax 9y 2x 3y 2x 3y It may also happen that the squares are not just simple monomial squares These next three examples illustrate such polynomials x 3 y x 3
21. 10x 8 In this example we are claiming that x 2 x 3x 4 x 5x 10x 8 for all real numbers In addition to going back over our work how can we verify such a claim Obviously we cannot try all real numbers but trying at least one number gives us a partial check Let s try the number 4 x 2 x 3x 4 4 2 4 3 4 4 2 16 12 4 2 8 16 x 5x 10x 8 43 5 4 10 4 8 64 80 40 8 16 We can also use a graphical approach as a partial check for such a problem In Fig ure 0 18 we let Y x 2 x 3x 4 and Y x 5x 10x 8 and graphed them on the same set of axes Note that the graphs appear to be identical 10 15 13 Figure 0 18 REMARK Graphing on the Cartesian coordinate system is not formally reviewed in this text until Chapter 2 However we feel confident that your knowledge of this topic from previous mathematics courses is sufficient for what we are doing at this time Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 32 Chapter O Some Basic Concepts of Algebra A Review Exponents can also be used to indicate repeated multiplication of polynomi als For example 3x 4y means 3x 4y 3x 4y and x 4 means x 4 x 4 x 4 Therefore raising a polynomial to a power
22. 42i e 1 3iV2 84 a 2 iV3 c iV2 2 7 24i 10 3i V5 4 Chapter 0 Review Problem Set page 86 1d 2 1 3 16 4l s g 125 7 8 9 7 3 6 9 2 3 W18 2 a 382 qx 45 12 V3 2 3x 47 3V7 49 1v3 _89V2 6 30 53 48 V 6 55 10 6 8V 30 57 3x V6y 6V2xy 59 13 7 V3 61 30 11 V6 63 16 65 x 2V xy y 67 a b 69 3V5 6 Th VI V3 2V10 3V14 x Vx 73 75 43 x 1 7 x V xy 99 6x 7V xy 2y x 9x 4y 2 81 V2x 2h NV 2x 1 83 Vx h 3 Vx 3 Problem Set 0 7 page 74 1 7 3 8 5 4 7 2 9 64 1 11 0 001 13 36 15 2 17 15x7 19 y 21 64x y 23 4x 5 7 16x43 y UEM LM 2 2 25 RI Bly s 31 8a x 33 V 8 35 Vx 37 xy V xy 39 a Va b 41 4 2 43 V2 45 V2 47 xV x 5N y 3 y KY 15 8 ERES sIN A xw x y y sv 9 8 55 EM 57 a V 2 c Vx X 2x 2 x 4x 6l 63 65 2x 1 x 2xy 2x 69 a 13 391 c 2 702 e 4 304 Problem Set 0 8 page 82 1 13 8i 3 3 4i amp Yi ti 7 eqs db ao d 20 12 10 12 2 13 4 07 15 3i 17 iV 19 19 lt i 21 272 23 3i V3 25 3i V6 27 18i 29 12i V2 31 8 33 V6 35 21 5 37 2V15 39 2V 14 41 3 43 V 6 45 21 07 47 8 12i 49 0 26i 51 53 26i 53 10 24i 55 14 8i 57 7 24i 59 3 47 61 113 07 63 13 Oi 65 ctu Ld le 7 13 13 m 2 12 16 20 23 25 27 1 8 9 4
23. 5 xy 5x 2y 10 In the next example notice that each term of the first polynomial multiplies each term of the second polynomial x 3 y z4 3 x y z 4 3 3 y z 4 3 xy xz 3x 3y 3z 9 Frequently multiplying polynomials produces similar terms that can be combined which simplifies the resulting polynomial x 5 x 7 x x 7 5 x 7 x Tx 5x 35 x 12x 35 In a previous algebra course you may have developed a shortcut for multi plying binomials as illustrated by Figure 0 17 o 2 2x 5 3x 2 2 6x2 10x 10 Figure 0 17 STEP 1 Multiply 2x 3x STEP 2 Multiply 5 3x and 2x 2 and combine STEP 3 Multiply 5 2 REMARK Shortcuts can be very helpful for certain manipulations in mathe matics But a word of caution Do not lose the understanding of what you are do ing Make sure that you are able to do the manipulation without the shortcut Keep in mind that the shortcut illustrated in Figure 0 17 applies only to mul tiplying two binomials The next example applies the distributive property to find the product of a binomial and a trinomial Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 0 3 Polynomials 31 x 2 x 3x 4 x x 3x 4 2 x 3x 4 x 3x 4x 2x 6x 8 x 5x
24. 5 V9 V5 3V5 V52 V4 13 V4V13 2V13 V24 V8 3 V8V3 2N 3 A variation of the technique for changing radicals with index n to simplest form is to factor the radicand into primes and then to look for the perfect nth pow ers in exponential form as in the following examples V80 V2 5 2S 2V5 4V5 W108 YZ 3 N 3N 2 34 The distributive property can be used to combine radicals that have the same index and the same radicand 3V2 5V2 3 5 V2 8V2 TVS 3N 5 7 3 N 5 405 Sometimes it is necessary to simplify the radicals first and then to combine them by applying the distributive property 3V8 2V18 4V2 3V4V2 2V9 V2 4V2 6V2 2 4V2 6 6 4 V2 8V2 Property 0 3 can also be viewed as WbWc Vbc Then along with the com mutative and associative properties of the real numbers it provides the basis for multiplying radicals that have the same index Consider the following two examples 7V6 3V8 7 3 V6 V8 21V48 21V16V3 21 4 V3 84V3 2 6 6 4 2 5 W6 W4 10V24 10V8W3 10 2 V3 2003 Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 64 Chapter O Some Basic Concepts of Algebra A Review The distributive property along with Property 0 3 provides a way of han dling special products involving radicals as the next examples illustrat
25. 5 Rational Expressions 59 2 3 2 4 x 3 x 3 x x 2 Te 5 2 3 3 x 9 x 3 x 2x x CE nd y 2 x x l 3 4 Ped 1 x xy 2x 2 S 8L 1 3 4 x b 3 TAE x 2 1 83 x P x 2 a 1 1 1 1 x h x 85 xth 1 x 1 h i h 3 _ 3 xth x h 2 p 2x 2h 1 2x 1 h 3 3 4x Ah t5 4x 5 h x y wrer ip x y Kay xt 2x7ly x 2y Ss 92 1 4x 3y ax Ey Explain in your own words how to multiply two ra tional expressions Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 60 Chapter O Some Basic Concepts of Algebra A Review 95 Give a step by step description of how to add 5 4 2 1 3x 5 8 9 4 14 5 2 Explain the reason for your choice of 96 Look back at the two approaches shown in Ex 14 21 ample 7 Which approach would you use to simplify approach for each problem 1 1 4 6 1 3 Which approach would you use to simplify 2 4 E Graphing calculator activities 97 Use the graphing feature of your graphing calcula 2x 3x 2 and 2x 1 tor to give visual support for your answers for Prob 2x 19x 5 4x 5 lems 60 68 d x3 2x 3x d x 98 For each of the following use your graphing calcu 33 46x2 5x 12 x 4 lator to help you decide whether the two given ex gt pressions are equivalent 5x lix 2 d z5x 1 an 3x 13x 14 3x 7 6x 7x 2
26. 66 71 75 73 34 109 30 000 111 0 03 75 1 77 11 79 4 117 a 4 385 10 c 2 322 10 e 3 052 10 Problem Set 0 2 page 26 rks Set Er page 35 14x x e x 4x 1 3 ES 5 27 7 4 5 6x T 4 0x bxc 7 9 x 34 2 16 11 12x3y 15x2y 13 30a P 24a D 18a b 9 E 11 1 13 25 15 4 15 x 20x 96 17 n 102 48 19 sx sy tx ty 21 6x 7x 3 17 or 0 01 19 or 0 00001 21 81 23 12x 37x 21 25 x 8x 16 100 100 000 27 Ar 12n 9 29 x x 14x 24 23 1 25 3 27 256 29 16 31 6x x Mx 6 33 x 2x 7x 4 16 4 25 25 35 7 1 37 6r x 5x 2 31 64 33 64 35 1 Gr0 00001 39 Y 8x3 hi 15x F 2x 4 41 25x 4 81 100 000 43 x 10x 21x 20x 4 45 4x 9y 683 Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 684 Problem Set 0 3 15x 75x 125 49 8x 12x 6x 1 64x 144x 108x 27 1253 150x y 60xy 8y 47 51 53 55 a 7a6b 21 b 35a D 35 b 21a D 4 Tab b 57 x 5x y 10x3y 10x2y 5xy y 59 x 8x7y 24x y 32xy 161 61 642 192a b 240a b 160a b 60a b 12ab b 63 x 7xPy 21xMy 35201 35x6y 21x y 4 7xiy 4 yl 65 324 240a
27. Inc All Rights Reserved amp oy 14 21 23 25 5 15 Xa id 38 5 a 3 x 6yY 2x 3y a a 2 yx 4y 3xy x9 8x 5 1 3 4 x 6 33 42x 12 Tx 35b 120 12 9n 10722 37 AA Ono 41 EEG 24 80a b 12n 9y 8x 12 43 DOr T Ay A 8 0 12xy 2x 1 3x 4 4 7x 21 49 1 5 5 x x 7 a 2 2 x 1 2n 10 il am 3 n 1 n 1 e x 1 9x 73 7 3 x 3 x 7 x 9 3x 30x 78 x 6 59 61 x 1 x 1 x 8 x 2 x 3 2 i re g gt 1 m 8 x 1 x 1 n 4 n 2 n 2 5x 16x 5 67 x 1 x 4 x 7 5 5 5y 3xy 69 a e eg 8 gj c x a 3 x y 4 2x 73 75 iu 6x 4 fa il n1 3x 9 xX x 1 d 40 1 2x h 79 1 lt x 1 4a t1 m xx hy 85 1 4 x 1 x h 1 2x 1 2x 2h 1 yx x 2 89 2 2 91 AS Aan xy xy 4y 3x Problem Set 0 6 page 68 1 9 3 5 5 gt 7 E 9 2V6 3V5 11 4V7 13 6 V11 15 n 9125 17 2x V3 19 8x2y3V y 21 23 4V 2 25 2x V2x 27 2x V 3x 2V3 V14 4V15 29 31 33 5 4 3 3V2 VAS 39 2150 1 ex Sab May not be copied scanned or duplicated in whole or in part Licensed to iChapters User Chapter 0 Review Problem Set 685 73 1 2 17 1 5 1 gt j 75 10 10 77 83 a 2 117 c 11
28. Negative real numbers do not have real nth roots For example there are no real fourth roots of 16 If n is an odd positive integer greater than 1 then the following statements are true 1 Every real number has exactly one real nth root 2 The real nth root of a positive number is positive For example the fifth root of 32 is 2 3 The real nth root of a negative number is negative For example the fifth root of 32 is 2 Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 62 Chapter O Some Basic Concepts of Algebra A Review In general the following definition is useful DEFINITION 0 5 Vo m ifandonly ifa 0 In Definition 0 5 if n is an even positive integer then a and b are both nonnegative If n is an odd positive integer greater than 1 then a and b are both nonnegative or both negative The symbol WV designates the principal root The following examples are applications of Definition 0 5 W81 3 because 34 81 V32 2 because 2 32 v 32 2 because 2 32 To complete our terminology the n in the radical Y b is called the index of the radical If n 2 we commonly write Vb instead of Wb In this text when we use symbols such as Wb Vy and Vx we will assume the previous agreements rel ative to the existence of real roots without listing the various restrictions unless a special restr
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30. a calculator and the square root key a V52 d V200 b V93 e V275 3 ER m o cA c Licensed to iChapters User 0 7 Relationship Between Exponents and Roots 71 92 The clamping process discussed in Problem 91 your calculator and the appropriate root keys to works for any whole number root greater than or check your answers equal to 2 For example a whole number approxi P A mation for V 80 is 4 because 4 64 and 5 125 a V24 b V 32 c V150 and 80 is closer to 64 than to 125 d 3200 e V 50 E W250 For each of the following use the clamping idea to find a whole number approximation Then use 0 7 Relationship Between Exponents and Roots Recall that we used the basic properties of positive integral exponents to motivate a definition of negative integers as exponents In this section we shall use the prop erties of integral exponents to motivate definitions for rational numbers as expo nents These definitions will tie together the concepts of exponent and root Let s consider the following comparisons From our study of If b b is to hold when m is a rational radicals we know number of the form 1 p where p is a positive that integer greater than 1 and n p then V 5y 5 51722 521 2 51 5 8 8 gry 930 3 gl 8 ad X24 g yap Ap 351 Such examples motivate the following definition DEFINITION 0 6 If b is a real number n is a positive integer greater than 1 and Vb exists
31. an easy to remember pattern 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 3 1 Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 34 Chapter O Some Basic Concepts of Algebra A Review Row number n in the formation contains the coefficients of the expansion of a b For example the fifth row contains1 5 10 10 5 1 800 these num bers are the coefficients of the terms in the expansion of a b Furthermore each can be formed from the previous row as follows 1 Start and end each row with 1 2 All other entries result from adding the two numbers in the row immedi ately above one number to the left and one number to the right Thus from row 5 we can form row 6 1 5 10 10 2 1 Add Add Add Add Add Row6 1 6 15 20 15 6 1 Row 5 Now we can use these seven coefficients and our discussion about the exponents to write out the expansion for a b a b a6 6a b 15a b 20a b 15a b 6ab b REMARK The triangular formation of numbers that we have been discussing is often referred to as Pascal s triangle This is in honor of Blaise Pascal a 17th cen tury mathematician to whom the discovery of this pattern is attributed Let s consider two more examples using Pascal s triangle and the exponent relationships Expand a By Solution a We can treat a b as a b and use the fourth row of Pasca
32. b 720a b 1080a7b 810ab 243p 67 3x 5x 69 5a 4a 9a 71 Sab 11a b 73 x y 75 x 3x 28 77 6x x 2 79 x 2x 1 81 x 6x 12x 8 Problem Set 0 4 page 46 1 2xy 3 4y 3 5 x y 3 a 9 3x 5 3x 5 13 z 3 x y 7 x ya b 11 1 9n 1 9n x 4 y x 4 y 15 19 23 27 29 33 3s 2t 1 3s 5 x 3 x 3x 5 x 2 x x 3Y x 3 2t 1 17 x 7 x 2 21 Not factorable 25 5x 1 2x 7 x 2 x 2x 4 4x 3y 16x 12xy 9y 31 4 x 4 35 3a 77 37 2n r 3n 5 39 5x 3 2x 9 41 6a 1y 43 4x y 2x y 45 Not factorable 47 2n n 7n 10 49 4 x 2 x 2x 4 51 x 3Y x 3 x 5 53 2y x 4 x 4 x 3 55 a b ct d a b c d 57 x 4 y x 4 y 59 x y x y 5 61 10x 3 6x 5 63 3x 7x 4 4x 5 65 x A x 4 67 x y x d Iy 3 y 69 x A x 7 71 2x 5 x 6 73 x y yx y y 75 a x 32 x 3 c x 21 x 24 e x 28 x 32 Problem Set 0 5 page 57 2x Ty amp x y a 4 1 i E Ts 3 9x i 9 a 9 x 2x 7 x 0 y 2 oy x 9 x 2y x 1 2x 9x 6xy 4y so fo 09 o x y x 5 2y Copyright 2005 Thomson Learning
33. be simplified to i 1 i or 1 as follows P QYyG 0 i i PPP yal 1 i iP 00 0 i 728 i y ay 1 Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 84 83 Chapter O Some Basic Concepts of Algebra A Review Express each of the following as i 1 i or 1 a i gt b 7 c iH d i e il fo g i h i We can use the information from Problem 82 and the binomial expansion patterns to find powers of complex numbers as follows 3 27 3P 3 3Y 2i 3 3 2iY 27 27 54i 367 8i 27 54i 36 1 8 i 9 46i Find the indicated power of each expression a 2 iy b 1 7 c 1 27 d 1 i e 2 7 f 1 7 Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part 84 Some of the solution sets for quadratic equations in the next chapter will contain complex numbers such as 4 V 12 2 and 4 V 12 2 We can simplify the first number as follows 4 v 12 4 iVv12 2 2 4 273 2 2 iV3 nies noa 2 73 2 2 Simplify each of the following complex numbers 4 V 12 6 V 24 be 2 4 1 V 18 6 V 27 o d 2 3 10 V 45 t 4 NV 48 e 4 2 CHAPTER 0 Licensed to iChapters Use
34. d 3x 2 a an 8x 6x 5 4x t5 4x 15x 54 x 6 and Ax 13x 9 x 1 0 6 Radicals Recall from our work with exponents that to square a number means to raise it to the second power that is to use the number as a factor twice For example 4 4 4 16 and 4 4 4 16 A square root of a number is one of its two equal factors Thus 4 and 4 are both square roots of 16 In general a is a square root of b if a b The following statements generalize these ideas 1 Every positive real number has two square roots one is positive and the other is negative They are opposites of each other 2 Negative real numbers have no real number square roots because the square of any nonzero real number is positive 3 The square root of zero is zero The symbol V called a radical sign is used to designate the nonnegative square root which is called the principal square root The number under the radical sign is called the radicand and the entire expression such as V 16 is referred to as a radical Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 0 6 Radicals 61 The following examples demonstrate the use of the square root notation V16 4 V 16 indicates the nonnegative or principal square root of 16 V16 4 V 16 indicates the negative square root of 16
35. examples which illustrate the process of finding a value of an algebraic expression The process is commonly referred to as evaluat ing an algebraic expression Find the value of 3xy 4z when x 2 y 4 and z 5 Solution 3xy 4z 3 2 4 4 5 when x 2 y 4 and z 5 24 20 4 m Find the value of a 4b 2c 1 when a 8 b 7 and c 14 Solution a 4b 2c 1 8 4 7 2 14 1 8 28 29 8 57 49 Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 14 Chapter O Some Basic Concepts of Algebra A Review 2b EEEIEE Evaluate L when a 14 b 12 c 3 and d 2 3c 5d Solution a 2b 14 2 12 3c 5d 3 3 5 2 14424 9 10 38 2 19 E Look back at Examples 1 3 and note that we use the following order of operations when simplifying numerical expressions You should also realize that first simplifying by combining similar terms can sometimes aid in the process of evaluating algebraic expressions The last example of this section illustrates this idea Evaluate 2 3x 1 3 4x 3 when x 5 Solution ts 23 2 3x 1 3 4x 3 2 3x 2 1 3 4x 3 3 6x 2 12x T9 6x t 11 Now substituting 5 for x w
36. fac torization of 288 288 2 3 Now we need to use five 2s and two 3s in the state ment TC j a4 Because 34 is divisible by 2 but not by 4 four factors of 2 must be in one number and one factor of 2 in the other number Also because 34 is not divisible by 3 both factors of 3 must be in the same number These facts aid us in determining that 2 2 2 2 3 3 34 or 16 18 Thus we can complete the original factoring prob lem x 34x 288 x 16 x 18 34 Use this approach to factor each of the following ex pressions a x 35x 96 b x 27x 176 c x 45x 504 d x 26x 168 e x 60x 896 f x 84x 1728 Consider the following approach to factoring 12x 54x 60 12x 54x 60 3x 6 4x 10 3 x 2 2 2x 5 6 x 2 2x 5 Is this factoring process correct What can you sug gest to the person who used this approach Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 48 Chapter O Some Basic Concepts of Algebra A Review 0 5 Rational Expressions Indicated quotients of algebraic expressions are called algebraic fractions or frac tional expressions The indicated quotient of two polynomials is called a rational expression This is analogous to defining a rational number as the indicated quo tient of two integers
37. iChapters User 80 Chapter O Some Basic Concepts of Algebra A Review Because complex numbers have a binomial form we can find the product of two complex numbers in the same way that we find the product of two binomials Then by replacing i with 1 we can simplify and express the final product in the standard form of a complex number Consider the following examples 2 3i 4 5i 2 4 5i 3i 4 5i 8 10i 12i 15 8 22i 15 1 8 22i 15 7 22i 1 7iY 1 72 1 7i 1 1 7i 7i 1 7i 1 7i 7i 497 1 14i 49 1 1 14i 49 48 14i 2 3i 2 3i 2 2 3i 3i 2 3i 4 6i 6i 97 4 9 1 449 13 REMARK Don t forget that when multiplying complex numbers we can also use the multiplication patterns a by a 2ab b a by a 2ab b a b a b a b The last example illustrates an important idea The complex numbers 2 3i and 2 3i are called conjugates of each other In general the two complex num bers a bi and a bi are called conjugates of each other and the product of a complex number and its conjugate is a real number This can be shown as follows a bi a bi a a bi bi a bi 0 abi abi bf g b 1 0 b Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or dup
38. in whole or in part Licensed to iChapters User 686 Chapter 0 Review Problem Set 70 V 71 2X 72 xVxy 12 1 73 Y 5 74 75 11 6i X 76 1 27 77 1 27 78 21 07 79 26 77 80 25 15i 81 14 12i 5 6 17 29 07 0 gt i y eed 82 29 Oi 83 0 gt i 84 53 260 12 30 0 i i 10i 85 0 7 oS ad 87 10i 88 2i V 10 89 16i V 5 90 12 91 4V3 92 2V2 93 600 000 000 94 800 000 Chapter 0 Test page 89 1 8 8 3 15 1 a A E 2 gp der x de xy 3 12x 8 4 30x 32x 8 5 3x 4x 11x 14 6 64x 48x 12x 1 7 9x y 12x y 8 3x 2x 1 3x 4 9 5x 2 6x 5 10 8 x 2 x 2x 4 21x x 3 11 x 2 x 12 Doe pe dae 20 2x4 72 8 23x 6 8 13n 14 15 16 12 2 6x x 3 x 2 s 2n 2y 5x 5V2 y 2 79 8 12V TRE 3y 4x 6 4V3 32 20 Aae 21 2xy V 6xy 22 4 3i 23 34 18i 24 85 07 25 42 le 14 L e 10 10 Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Photo Credits Chapter 0 Photodisc 1 Chapter 1 Photodisc 1 Chapter 2 Photodisc V Chapter 3 Photodisc V Chapter 4 Photodisc 18 Chapter 5 Photodisc 1 Chapter 6 Photodisc 1 Chapter 7 Photodisc 1 Chapter 8 Photodisc 1 Chapter 9 Photodisc 1 Chapter 70 Getty Images Copyright 2005
39. j p E 0 0064 420 000 8600 0 0000064 89 4 V 80 90 V 9 V 16 0 00014 0 032 0 0016 0 000043 91 V 6 V 8 ye V 3 Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part CHAPTER 0 Licensed to iChapters User TEST applies to all problems in this Chapter Test 1 Evaluate each of the following 3 b 5 4 3 2 227 C d 64 2 Find the product 3x y 5x y and express the result using positive exponents only a 7 For Problems 3 7 perform the indicated operations b 3k 4 he S ak 22 9 4 5x 2 6x 4 5 x 2 3x 2x 7 6 4x 17 18x y3 24x y xy For Problems 8 11 factor each polynomial completely Rb Je 15 2x 9 309 136 10 10 8x 64 T se 4 say m Dyp 25 For Problems 12 16 perform the indicated operations involving rational expressions Express final answers in simplest form 6xiy _ 8 ST ChapterO Test 89 2x 7x 3 x 8 x 4 BIS 25v sp Sue ap 2 Am 2 Ama l 4 6 E 4 15 2 SE ar uS 14 2x 6x 16 gt gt amp to CA 17 Simplify the complex fraction amp e For Problems 18 21 express each radical expression in simplest radical form All variables represent positive real numbers 18 6V28x V6 7 BV For Problems 22 25 perform the indica
40. nonzero real number then DA Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 20 Chapter O Some Basic Concepts of Algebra A Review Therefore according to Definition 0 3 the following statements are all true 50 21 413 1 0 1 ey 1 ifx 0and y 0 A similar line of reasoning can be used to motivate a definition for the use of negative integers as exponents Consider the example x x If b p p is to hold then x x should equal x 9 which equals x 1 Therefore x must be the reciprocal of x because their product is 1 That is x 1 x This suggests the following definition DEFINITION 0 4 If n is a positive integer and b is a nonzero real number then 1 p pu According to Definition 0 4 the following statements are true 1 1 1 MN 24 ug 2 16 3 1 d 106 _ 8 4 4 pE 49 Q7 x 47 16 The first four parts of Property 0 1 hold true for all integers Furthermore we do not need all three equations in part 5 of Property 0 1 The first equation b Ln m b can be used for all integral exponents Let s restate Property 0 1 as it pertains to in tegers We will include name tags for easy reference PROPERTY 0 2 If m and n are integers and a and b are real numbers with b 0 when ever it appears in a denominator then 1 b b b
41. we can write A 7 B which is read set A is not equal to set B Real Numbers The following terminology is commonly used to classify different types of numbers 1 2 3 4 Natural numbers counting numbers positive integers 0 1 2 3 Whole numbers nonnegative integers 4773 722 11 Negative integers 3 2 1 0 Nonpositive integers 72 1 0 1 2 Integers A rational number is defined as any number that can be expressed in the form a b where a and b are integers and b is not zero The following are examples of rational numbers 2 3 1 1 13 r 65 because 65 gt 3 4 6 4 4 6 because 6 7 4 because ique 0 b TL t 0 3 b 0 3 ecause 0 a um z ete ecause 0 3 10 A rational number can also be defined in terms of a decimal representation Before doing so let s briefly review the different possibilities for decimal repre sentations Decimals can be classified as terminating repeating or nonrepeating Here are some examples of each 0 3 0 46 Terminating decimals 0 789 0 2143 0 3333 0 141414 0 712712712 Repeating decimals 0 24171717 0 9675283283283 Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 4 Chapter O Some Basic Concepts of Algebra A Review Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated
42. we enter 589 and press the ENTER key the display will show 5 89E2 Likewise when the calculator is in scientific mode the answers to compu tational problems are given in scientific form For example the answer for 76 533 is given as 4 0508E4 It should be evident from this brief discussion that even when you are using a calculator you need to have a thorough understanding of scientific notation Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 26 Chapter O Some Basic Concepts of Algebra A Review PROBLEM SET 0 2 For Problems 1 42 evaluate each numerical expression 1 2 2 37 3 10 4 10 1 1 5 33 6 25 1 2 1 2 2 s 3 INES 5 3 w 8 1 1 11 12 GY 5 1 4 5 15 2 2 16 32 3 17 10 5 10 18 106 10 19 10 10 20 10 10 21 32 22 2 1 3 23 4 24 371 25 37 x 2 26 23 3 77 27 STY 28 27 4 22 2 22 32 2 v s 371 2 5 2 N21 a 8 2 273 33 34 Sr 107 107 35 s 36 05 371 3242 38 2 57 2 3N 9 2 5 2 40 3 2 23 41 Q 3717 42 32 511 Simplify Problems 43 62 express final results without using zero or negative integers as exponents 43 x3 x77 44 x x 2 45 a a 46 b 2 b5 b 47 a Oy 48 b 49 xy 50 xy
43. x 6x 8 3 x 4 x 2 3x 3y 3 x y 3 x y x xy y a b a b a b a b y a b a b x 6x 27 x 9 x 3 x 3 x 3 x 3 3x4y 9x y 84y 3y x 3x 28 3y x T x 4 3y x T x 2 x 2 x y 8y 16 x y 8y 16 ty 4 x y 49 x y 4 gees E PROBLEM SET 0 4 Factor each polynomial completely Indicate any that 19 15 2x x 20 40 6x x are not factorable using integers J 5 21 x 7x 36 22 x Axy Sy 1 6xy 8xy 2 4a b 12ab 23 3x 11x 10 24 2x Ix 30 Sex RS Tue OT ET ag Ai ede s 26 872 227 21 5 3x 3y ax ay 6 ac be at b 27 3 8 28 x3 64 2 7 ax ay bx by 8 2a 300 200 300 20 64x3 27y 30 27x 8y 9 9x 25 10 4x7 9 31 4x 16 32 n 49n 11 1 81n 12 9x y 64 33 x 9x 34 12772 59n 72 13 x 4 y 14 x y 1 35 9a 42a 49 36 1 16x 15 957 Qr 1y 16 4a 3b 1 37 2n 6n 10n 38 x y 7 17 x 5x 14 18 a 5a 24 39 10x 39x 27 40 x 5 Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 41 43 45 47 49 51 53 55 56 57 59 6l 63 36a
44. 1 91 x x 92 2y9 4y 1 3a Ax 94 x 2 2 93 EH Thoughts into words 113 Explain how you would simplify 3 2 and also how you would simplify 3 2 0 2 Exponents 27 24y 1 95 60 96 x y x b 2x A4xP 1 y G2 p QC gx For Problems 99 102 express each number in scientific notation 99 62 000 000 101 0 000412 100 17 000 000 000 102 0 000000078 For Problems 103 106 change each number from sci entific notation to ordinary decimal form 103 1 8 10 104 5 41 10 105 2 3 10 106 4 13 10 For Problems 107 112 use scientific notation and the properties of exponents to help perform the indicated operations 0 00052 0 013 109 V 900 000 000 110 0 000004 0 00069 0 0034 0 0000017 0 023 0 000075 4 800 000 15 000 0 0012 107 111 V 0 0009 114 How would you explain why the product of x and x is x and not x a Graphing calculator activities 115 Use your calculator to check your answers for Problems 107 112 116 Use your calculator to evaluate each of the follow ing Express final answers in ordinary notation a 27 000 b 450 000 c 14 800 e 900 d 1700 f 60 g 0 0213 h 0 000213 i 0 000198 j 0 000009 Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to
45. 3 Now let s consider an example in which the denominator is of binomial form Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User EXAMPLE 4 EXAMPLE 5 em 0 6 Radicals 67 4 Simplify by rationalizing the denominator V5 V2 Solution Remember that a moment ago we found that V5 V2 V5 V2 3 Let s use that idea here 4 4 A 8 V2 NA S V2 v5 e v2 amp V5 V2 4 V5 V2 2 Vs NVAINS V2 3 The process of rationalizing the denominator does agree with the previously listed conditions However for certain problems in calculus it is necessary to rationalize the numerator Again the fact that Va Vb Va Vb a b can be used Vx h Vx Change the form of h by rationalizing the numerator Solution Ma _ cxt h vs h h m PEVA _ xth x h V x h 4 Vx h OK V x ho Vx 1 Ned do Vx Radicals Containing Variables Before we illustrate how to simplify radicals that contain variables there is one im portant point we should call to your attention Let s look at some examples to il lustrate the idea Consider the radical V x for different values of x Let x 3 then Vx V3 V9 3 Let x 3 then Vx V 3 79 3 Copyright 2005 Thomson Learning Inc All Rights Reserved May
46. 32 a 16 b 257 c V 2401 d W 65 536 c 16 d 275 e V 161 051 f 6 436 343 e 3437 3 f 51243 n 67 In Definition 0 7 we stated that b Vb Vby Use your calculator to verify each of the following 69 Use your calculator to estimate each of the follow ing to the nearest thousandth a 74 3 b 1045 3 93 Jg N a V2 V27p b V S N 8y e 1225 d 1975 4 M y e V16 W165 a W162 W16 e TA f 10 8 Viz4 232 0 8 Complex Numbers So far we have dealt only with real numbers However as we get ready to solve equations in the next chapter there is a need for more numbers There are some very simple equations that do not have solutions if we restrict ourselves to the set of real numbers For example the equation x 1 0 has no solutions among the real numbers To solve such equations we need to extend the real number system In this section we will introduce a set of numbers that contains some numbers with squares that are negative real numbers Then in the next chapter and in Chapter 5 we will see that this set of numbers called the complex numbers provides solutions not only for equations such as x 1 0 but also for any polynomial equation in general Let s begin by defining a number i such that i 2 1 Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 0 8 Complex Numbers 77
47. 3a7 12 a 16 3a 12 s 3a 10 3a 150 a 30 10 3a2 15a a 16 9 2 a2 4 a 35 a 4 2 3a a S a2 4 a 2 a 2 u 1 a a 2 Adding and Subtracting Rational Expressions The following two properties provide the basis for adding and subtracting rational expressions Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User EXAMPLE 1 0 5 Rational Expressions 51 a z c a qu b b b S 26i He b b b These properties state that rational expressions with a common denominator can be added or subtracted by adding or subtracting the numerators and placing the result over the common denominator Let s illustrate this idea 8 3 80 3 11 x 2 x 2 x 2 x 2 9 7 9 7 2 1 4y 4y 4y 4y 2y Don t forget to simplify the final result n 1 n 1 n F 1 n n 1 n 1 72 1 n t If we need to add or subtract rational expressions that do not have a common denominator then we apply the property a b a k b k to obtain equivalent fractions with a common denominator Study the next examples and again pay spe cial attention to the format we used to organize our work REMARK Remember that the least common multiple of a set of whole numbers is the smallest nonzero whole number divisible by each of the numbers in the set When we add or subtract rational numbers the least common
48. 5 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 0 4 Factoring Polynomials 39 6x y 27xy 3xy3 2x 9y 30x 42x4 24x 6x 5 7x Ax Sometimes there may be a common binomial factor rather than a com mon monomial factor For example each of the two terms in the expression x y 2 z y 2 has a binomial factor of y 2 Thus we can factor y 2 from each term and obtain the following result x y 2 z y 2 y 2 x z Consider a few more examples involving a common binomial factor a b 1 2 b 1 b 1 a 2 x 2y 1 yy 1 Qy D x y x x 2 3 x 2 x 2 x 3 It may seem that a given polynomial exhibits no apparent common monomial or binomial factor Such is the case with ab 3c bc 3a However by using the commutative property to rearrange the terms we can factor it as follows ab 3c bc 3a ab 3a bc 3c a b 3 c b 3 Factor a from the first two terms and c from the last two terms b 3 at c Factor b 3 from both terms This factoring process is referred to as factoring by grouping Let s consider an other example of this type ab 4b 3a 12 b a 4 3 a 4 Factor b from the first two terms 3 from the last two a 4 D 3 Factor the common binomial from both terms Difference of Two Squares In Section
49. 6 8 5 W 8 2 16 It can be shown that all of the results pertaining to integral exponents listed in Property 0 2 on page 20 also hold for all rational exponents Let s consider some examples to illustrate each of those results xi xP x 1 2 2 3 b b pn m y3 6 4 6 1 6 X a2 32 a 9 00 5 b y pnm a a 16282 16 2 y25512 ab ap ES 4y i 3 4 n yt ye b pnm 1 2 p y ye y uns quay aye y 71300 b bp x y Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User EXAMPLE 1 0 7 Relationship Between Exponents and Roots 73 The link between exponents and roots provides a basis for multiplying and divid ing some radicals even if they have different indexes The general procedure is to change from radical to exponential form apply the properties of exponents and then change back to radical form Let s apply these procedures in the next three examples 3 5 91 2 91 3 91 2 1 3 95 6 6 55 16 202 21 20 2 2 Faya Vxy V xy xy x y 1 2 2 5 ys 1242 5 y1 2 1 5 x 2y x 10y7 10 10 xy yum UN xy 1 2 1 3 51 6 405 WS dp ES Earlier we agreed that a radical such as N x is not in simplest form because the radicand contains a perfect power of the index Thus we simplified WA by ex pressing it as We Wx which in tur
50. COLLEGE ALGEBRA SIXTH AL TX s prai TE El t TIT aS to ACA N v E KAUFMAN JEROM College Algebra JEROME E KAUFMANN KAREN L SCHWITTERS Seminole Community College THOMSON BROOKS COLE Australia e Canada Mexico e Singapore e Spain United Kingdom United States Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User THOMSON BROOKS COLE Acquisitions Editor John Paul Ramin Assistant Editor Katherine Brayton Editorial Assistant Darlene Amidon Brent Technology Project Manager Rachael Sturgeon Marketing Manager Karin Sandberg Marketing Assistant Erin Mitchell Advertising Project Manager Bryan Vann Project Manager Editorial Production Belinda Krohmer COPYRIGHT 2005 Thomson Brooks Cole a part of The Thomson Corporation Thomson the Star logo and Brook Cole are trademarks used herein under license ALL RIGHTS RESERVED No part of this work covered by the copyright hereon may be reproduced or used in any form or by any means graphic electronic or mechanical including photocopying recording taping Web distribution information storage and retrieval systems or in any other manner without the written permission of the publisher Printed in the United States of America 12345 67 09 08 07 06 05 For more information about ou
51. Oy y Wy 82 6x y i 6 x yt axy i 3y Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 0 5 Rational Expressions 49 The factoring techniques discussed in the previous section can be used to fac tor numerators and denominators so that the property a k b k a b can be applied Consider the following examples x Ax x x 4 x x 16 x 4 x 4 x 4 5m 6n 8 _ SH 4 n 2 n 2 10n 3n 4 5ua 4 Qn 1 2n 1 y x yy x xy y x 4 xy 2x 42y x x y 2 x y equa xy y 2a y x 2 4742 6x y 6xy 6xy x 1 6xy x 1 x 1 6y x 1 x Sx 4x x xX 5x44 x x2 x 4 x44 Note that in the last example we left the numerator of the final fraction in fac tored form This is often done if expressions other than monomials are involved Either 6y x 1 or 6xy 6y x 4 x 4 is an acceptable answer Remember that the quotient of any nonzero real number and its opposite is 1 For example 6 6 1 and 8 8 1 Likewise the indicated quotient of any polynomial and its opposite is equal to 1 For example 1 because a and a are opposites a a b b 1 because a b and b a are opposites a x 4 5 1 gl because x 4 and 4 x are opposites mE The next example illustrates how we us
52. Product of two powers Pas 104 em 0 Power of a power 3 ab a b Power of a product Power of a quotient Quotient of two powers Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User EXAMPLE 5 0 2 Exponents 21 Having the use of all integers as exponents allows us to work with a large va riety of numerical and algebraic expressions Let s consider some examples that il lustrate the various parts of Property 0 2 Evaluate each of the following numerical expressions 279 2 a 21 33 b os Solutions a 271 33 ur Power of a product 253 Power of a power e 2733 2 b Power of a quotient Power of a power _ 64 mi EXAMPLE 6 Find the indicated products and quotients and express the final results with posi tive integral exponents only ds a 3x y 4Ax y b Me c S Solutions a 3z2y S 4x Ay 123 C9 Product of powers 12x y _2 xy A EA 404 090p Quotient of powers 4atb NN x Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 22 Chapter O Some Basic Concepts of Algebra A Review 15x y N 71 c 5 x egit First simplify inside parentheses Xy 3x 3y 3 tyy Power of a product x E The next two examples illustrate the simplificati
53. The following are examples of rational expressions 3 x 2 x 5x 1 xy x y a 302 5a 1 5 x3 x 9 xy a ta 6 Because division by zero must be avoided no values can be assigned to variables x 2 x b meaningful for all real number values of x except x 3 Rather than making re strictions for each individual expression we will merely assume that all denomina tors represent nonzero real numbers The basic properties of the real numbers can be used for working with ratio nal expressions For example the property that will create a denominator of zero Thus the rational expression ak a b k b which is used to reduce rational numbers is also used to simplify rational expres sions Consider the following examples 15x 3px y 3x sys 5 5 1 9 9 1 18x y 18x y 27 2 Note that slightly different formats were used in these two examples In the first one we factored the coefficients into primes and then proceeded to simplify how ever in the second problem we simply divided a common factor of 9 out of both the numerator and denominator This is basically a format issue and depends upon your personal preference Also notice that in the second example we applied the property This is part of the general property that states The properties b b and ab a b may also play a role when sim plifying a rational expression as the next example demonstrates 8 xj AxXiyy 4
54. V 9 10 Vaio 3 10 3 100 300 E Many calculators are equipped to display numbers in scientific notation The display panel shows the number between 1 and 10 and the appropriate exponent of 10 For example evaluating 3 800 000 yields 1 444E13 Thus 3 800 000 1 444 10 14 440 000 000 000 Similarly the answer for 0 000168 is displayed as 2 8224E 8 Thus 0 000168 2 8224 10 0 000000028224 Calculators vary in the number of digits they display between 1 and 10 when they represent a number in scientific notation For example we used two different calculators to estimate 6729 and obtained the following results 9 283316768E22 9 28331676776E22 Obviously you need to know the capabilities of your calculator when working with problems in scientific notation Many calculators also allow you to enter a number in scientific notation Such calculators are equipped with an enter the exponent key often labeled EE Thus a number such as 3 14 10 might be entered as follows Sud EE 3 14E 8 3 14E8 A MODE key is often used on calculators to let you choose normal decimal notation scientific notation or engineering notation The abbreviations Norm Sci and Eng are commonly used If the calculator is in scientific mode then a number can be entered and changed to scientific form with the ENTER key For example when
55. V0 0 Zero has only one square root Technically we could also write V0 0 0 V 4 Not a real number V 4 Not a real number To cube a number means to raise it to the third power that is to use the number as a factor three times For example 2 2 2 2 8 and 2 2 2 2 8 A cube root of a number is one of its three equal factors Thus 2 is a cube root of 8 and as we will discuss later it is the only real number that is a cube root of 8 Furthermore 2 is the only real number that is a cube root of 8 In general a is a cube root of b if b The following statements generalize these ideas 1 Every positive real number has one positive real number cube root 2 Every negative real number has one negative real number cube root 3 The cube root of zero is zero REMARK Every nonzero real number has three cube roots but only one of them is a real number The other roots are complex numbers which we will discuss in Section 0 8 The symbol V isusedto designate the cube root of a number Thus we can write 3 3 gist aq 1 8 2 W 8 2 v8 27 3 27 3 The concept of root can be extended to fourth roots fifth roots sixth roots and in general nth roots If n is an even positive integer then the following state ments are true 1 Every positive real number has exactly two real nth roots one positive and one negative For example the real fourth roots of 16 are 2 and 2 2
56. V4 2i V 17 iV17 V 24 iV24 iV4V6 2i V6 Note that we simplified the radical V24 to 2 V6 We should also observe that V b where b gt 0 is a square root of b because V b ivby iXb 1 b b Thus in the set of complex numbers b where b gt 0 has two square roots i Vb and i V b These are expressed as V b iVb and WV b iVb We must be careful with the use of the symbol V b where b gt 0 Some properties that are true in the set of real numbers involving the square root symbol do not hold ifthe square root symbol does not represent a real number For example Va Vb Vab does not hold if a and b are both negative numbers Correct V 4V 9 27 37 6i 6 1 6 Incorrect V 4N 9 V 4X 9 V36 6 To avoid difficulty with this idea you should rewrite all expressions of the form V b where b gt 0 in the form i Vb before doing any computations The following examples further illustrate this point V 5NV 1 iV Si VT i N 35 1 V35 V35 V 2V 8 iV2 i V8 75616 1 4 4 V 2N 8 iV2 V8 iV16 4i V 6V 8 iV 6i V8 i V48 i V16V3 4i V3 4V3 vV 2 iV2 ivV2 V3 iV6 V3 yB VBV 3 v 48 ivas J8 vga Vin N 2 iv4 2i Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to
57. adicals with different indexes 2 to change to simplest radical form while in exponential form and 3 to simplify expressions that are roots of roots A complex number is any number that can be expressed in the form a bi where a and b are real numbers and i is the imaginary unit such that i 1 Addition and subtraction of complex numbers are defined as follows Geter toy se rese e es er sec ae o Se eu ace bi er Gi e 9 8 b 8 Because complex numbers have a binomial form we can multiply two complex numbers in the same way that we multiply two binomials Thus i can be replaced with 1 and the final result can be expressed in the standard form of a complex number For example 3 27 4 31 212 2 677 12 i 6 1 18 i The two complex numbers a bi and a bi are called conjugates of each other The product a bi a bi equals the real number a b and this property is used to help with dividing complex numbers CHAPTER O REVIEW PROBLEM SET For Problems 1 10 evaluate 1 57 2 34 T7 5 5 V 64 6 y2 EV 5 8 36 12 9 10 32 5 For Problems 11 18 perform the indicated operations and simplify Express the final answers using positive ex ponents only 11 3x 2y 1 4x4y 12 5x 6x 2 13 8a 6a 14 3x 5y15y 64x y 36ry S6x 13 y E Ie Copyright 2005 Thomson Learning Inc All Rights Reserved 8 2
58. amples illustrate 3 x 2 3 x 3 2 3x 6 3x x 4 3x x 3x 4 3x 12x For factoring purposes the distributive property now in the form ab ac a b c can be used to reverse the process The steps indicated in the dashed boxes can be done mentally Polynomials can be factored in a variety of ways Consider some factoriza tions of 3x 12x 3x 12x 3x x 4 or 3x 12x 3 x 4x or 1 3x 12x x 3x 12 or 3x 12x 8 2660 24x We are however primarily interested in the first of these factorization forms we refer to it as the completely factored form A polynomial with integral coefficients is in completely factored form if l it is expressed as a product of polynomials with integral coefficients and 2 no polynomial other than a monomial within the factored form can be further factored into polynomials with integral coefficients Do you see why only the first of the factored forms of 3x 12x is said to be in com pletely factored form In each of the other three forms the polynomial inside the 1 parentheses can be factored further Moreover in the last form 3 6x 24x the condition of using only integers is violated This application of the distributive property is often referred to as factoring out the highest common monomial factor The following examples illustrate the process 12x 16x 4x 3x 4 8ab 18b 2b 4a 9 Copyright 200
59. ated in whole or in part The general nature of algebra makes it applicable to a large variety of occupations Photodisc he temperature in Big Lake Alaska at 3 P M was 4 F By 11 P M the temperature had dropped another 20 We can use the numerical expression 4 20 to determine the temperature at 11 P M Megan has p pennies n nickels d dimes and q quarters The algebraic expression p 5n 10d 250 can be used to repre sent the total amount of money in cents Algebra is often described as a generalized arithmetic That description does not tell the whole story but it does convey an important idea A good understanding of arithmetic provides a sound basis for the study of algebra In this chapter we will often use arithmetic examples to lead into a review of basic algebraic concepts Then we will use the algebraic concepts in a wide variety of problem solving situations Your study of alge bra should make you a better problem solver Be sure that you can work effectively with the algebraic concepts reviewed in this first chapter Copyright 2005 Thomson Learning Inc All Rights Reseryed May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 2 Chapter O Some Basic Concepts of Algebra A Review 0 1 Some Basic Ideas Let s begin by pulling together the basic tools we need for the study of algebra In 2 arithmetic symbols such as 6 3 0 27 and m are u
60. be used to factor trino mials Let s consider some examples Factor x 12x 20 Solution We need two integers whose sum is 12 and whose product is 20 The numbers are 2 and 10 and we can complete the factoring as follows x 12x 20 x 2 x 10 El Factor x 3x 54 Solution We need two integers whose sum is 3 and whose product is 54 The integers are 9 and 6 and we can factor as follows x 3x 54 x 9 x 6 Factor x 7x 16 Solution We need two integers whose sum is 7 and whose product is 16 The only possible pairs of factors of 16 are 1 16 2 8 and 4 4 A sum of 7 is not produced by any of these pairs so the polynomial x 7x 16 is not factorable using integers Trinomials of the Form ax bx c Now let s consider factoring trinomials where the coefficient of the squared term is not one First let s illustrate an informal trial and error technique that works well for certain types of trinomials This technique is based on our knowledge of multi plication of binomials Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 42 Chapter O Some Basic Concepts of Algebra A Review EXAMPLE 4 Factor 3x 5x 2 Solution By looking at the first term 3x and the positive signs of the other two terms we know that the binomials are of the form x
61. bi where b 0 a bi where b 0 Pure imaginary numbers a bi where a 0 and b 0 Two complex numbers a bi and c di are said to be equal if and only if a cand b d In other words two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 78 Chapter O Some Basic Concepts of Algebra A Review Adding and Subtracting Complex Numbers The following definition provides the basis for adding complex numbers a bi c di a t c b t d i We can use this definition to find the sum of two complex numbers 4 3i 5 9i 2 A 5 c 3 9i 2 9 12i 76 4i 8 7i 6 8 4 7i 22 3i 1 3 2 1 i 2 342 3734744 7 i3 4 PENCIL 3 Jj 5 by 7 19 T Led 6 6 20 20 6 20 3 iV2 4 iV2 8 4 V2 Vi 1 2i V2 Note the form for writing 2 V 2i The set of complex numbers is closed with respect to addition that is the sum of two complex numbers is a complex number Furthermore the commutative and associative properties of addition hold for all complex numbers The additive identity element is 0 Oi or simply the real number 0 The additive inverse of a biis a bi because a bi a bi a a b b i 0 Theref
62. ctor as follows 5x 18x 8 5x 20x 2x 8 5x x 4 2 x 4 x 4 5x 2 E Factor 24x 2x 15 Solution eL 24x 2x 15 Sum 072 Product of 24 15 360 We need two integers whose sum is 2 and whose product is 360 To help find these integers let s factor 360 into primes 360 2 2 2 3 3 5 Now by grouping these factors in various ways we find that 2 2 5 20 and 2 3 3 18 so we can use the integers 20 and 18 to produce a sum of 2 and a Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 0 4 Factoring Polynomials 45 product of 360 Therefore the middle term 2x of the trinomial can be expressed as 20x 18x and we can proceed as follows 24x 2x 15 24x 20x 18x 15 4x 6x 5 3 6x 5 6x 5 4x 3 a Probably the best way to check a factoring problem is to make sure the con ditions for a polynomial to be completely factored are satisfied and the product of the factors equals the given polynomial We can also give some visual support to a factoring problem by graphing the given polynomial and its completely factored form on the same set of axes as shown for Example 10 in Figure 0 19 Note that the graphs for Y 24x 2x 15 and Y 6x 5 4x 3 appear to be identical 20 20 Figure 0 19 Sum a
63. d not have a calculator or table available ra Graphing calculator activities 90 Sometimes it is more convenient to express a large or very small number as a product of a power of 10 and a number that is not between 1 and 10 For ex ample suppose that we want to calculate V 640 000 We can proceed as follows 640 000 V 64 10 64 10 64 10 2 8 10 8 100 800 Compute each of the following without a calculator and then use a calculator to check your answers a V 49 000 000 b V 0 0025 c V 14 400 d V 0 000121 e V 27 000 f 00 000064 Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part 91 There are several methods of approximating square roots without using a calculator One such method works on a clamping between values principle For example to find a whole number approximation for V 128 we can proceed as follows 11 121 and 12 144 Therefore 11 lt V 128 lt 12 Because 128 is closer to 121 than to 144 we say that 11 is a whole number approximation for V128 If a more precise approximation is needed we can do more clamping We would find that 11 3 127 69 and 11 4 129 96 Because 128 is closer to 127 69 than to 129 96 we conclude that V128 11 3 to the nearest tenth For each of the following use the clamping idea to find a whole number approximation Then check your answers using
64. dinate a of x x 0 2 Locate its opposite written as x b tt on the other side of zero x 0 3 Locate the opposite of x written CQ Sa as x on the other side of ZTO Figure 0 5 Therefore we conclude that the opposite of the opposite of any real number is the number itself and we express this symbolically by x x REMARK The symbol 1 can be read negative one the negative of one the oppo site of one or the additive inverse of one The opposite of and additive inverse of terminology is especially meaningful when working with variables For example the symbol x read the opposite of x or the additive inverse of x emphasizes an important issue Because x can be any real number x opposite of x can be zero positive or negative If x is positive then x is negative If x is negative then x is positive If x is zero then x is zero The concept of absolute value can be interpreted on the number line Geo metrically the absolute value of any real number is the distance between that num ber and zero on the number line For example the absolute value of 2 is 2 the ab solute value of 3 is 3 and the absolute value of zero is zero see Figure 0 6 I 3 23 21 2 A r k 3 2 1 0 1 2 3 0 0 Figure 0 6 Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated i
65. do serve a purpose at times The Use of Sets Some of the vocabulary and symbolism associated with the concept of sets can be effectively used in the study of algebra A set is a collection of objects the objects are called elements or members of the set The use of capital letters to name sets and the use of set braces to enclose the elements or a description of the ele ments provide a convenient way to communicate about sets For example a set A that consists of the vowels of the alphabet can be represented as follows A vowels of the alphabet Word description Or A fa e i 0 u List or roster description or A x x is a vowel Set builder notation Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 0 1 Some Basic Ideas 3 A set consisting of no elements is called the null set or empty set and is written Z Set builder notation combines the use of braces and the concept of a vari able For example x x is a vowel is read the set of all x such that x is a vowel Note that the vertical line is read such that Two sets are said to be equal if they contain exactly the same elements For example 1 2 3 2 1 3 because both sets contain exactly the same elements the order in which the elements are listed does not matter A slash mark through an equality symbol denotes not equal to Thus if A 1 2 3 and B 3 6
66. e 2V2 4V3 5V6 2V2 4V3 2vV2 5V6 8V6 10V12 8V6 10V4V3 8V6 20V3 2V2 V7 3V2 5V7 2V23V2 5V7 V7 3V2 5V7 2V2 3V 2 2V2 5V7 VIB V2 V7 5V7 6 2 10V14 3V14 5 7 23 7V14 V5 V2YV5 V2 VS V5 V2 V2 V 5 V2 VSYV5 V5 V2 V2YV5 V2Y V2 5 V10 V10 2 3 Pay special attention to the last example It fits the special product pattern a b a b a b We will use that idea in a moment More About Simplest Radical Form Another property of nth roots is motivated by the following examples 36 36 6 Vc and Va Saa vo 3 3 495 48 2 and i22 8 We 2 In general the following property can be stated PROPERTY 0 4 we if Wb and We are real numbers and c 0 c We Property 0 4 states that the nth root of a quotient is equal to the quotient of the nth roots 4 12 To evaluate radicals such as 25 and 4 where the numerator and the denominator of the fractional radicands are perfect nth powers we can either use Property 0 4 or rely on the definition of nth root Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 0 6 Radicals 65 14 774 2 14 RET NE B s 5 25 3 ecause 5 55 P en Or Tla TT 8 138 2 8 2 222 8 Radicals such as 4 s and 4 where only the denomina
67. e number For example 35 10 3 5 and 145 10 14 5 Now let s observe the following pattern when squaring such a number 10x 5 100x 100x 25 100x x 1 25 The pattern inside the dashed box can be stated as add 25 to the product of x x 1 and 100 Thus to compute 35 mentally we can think 35 3 4 100 25 1225 Compute each of the fol lowing numbers mentally and then check your an swers with your calculator a 15 b 25 e 45 d 55 e 65 f 75 g 85 h 95 i 105 If a polynomial is equal to the product of other polynomials then each polynomial in the product is called a factor of the original polynomial For example because x 4 can be expressed as x 2 x 2 we say that x 2 and x 2 are factors Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 38 Chapter O Some Basic Concepts of Algebra A Review of x 4 The process of expressing a polynomial as a product of polynomials is called factoring In this section we will consider methods of factoring polynomials with integer coefficients In general factoring is the reverse of multiplication so we can use our knowl edge of multiplication to help develop factoring techniques For example we pre viously used the distributive property to find the product of a monomial and a poly nomial as the next ex
68. e obtain 6x 11 6 5 11 30 11 41 One calculator method for evaluating an algebraic expression such as 3xy 4z for x 2 y 4 and z 5 see Example 1 is to replace x with 2 y with 4 Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 0 1 Some Basic Ideas 15 and z with 5 and then calculate the resulting numerical expression Another method is shown in Figure 0 14 in which the values for x y and z are stored and then the algebraic expression 3xy 4z is evaluated Figure 0 14 PROBLEM SET 0 1 For Problems 1 10 identify each statement as true or false 1 10 Every rational number is a real number Every irrational number is a real number Every real number is a rational number If a number is real then it is irrational Some irrational numbers are also rational numbers All integers are rational numbers The number zero is a rational number Zero is a positive integer Zero is a negative number All whole numbers are integers For Problems 11 18 list those elements of the set of numbers o V5 V2 Z ES 1L 0 279 0 467 ar 14 46 675 8 that belong to each of the following sets 11 The natural numbers 12 The whole numbers 13 The integers 14 The rational numbers 15 The irrational numbers 16 The nonnegative integers 17 The nonposit
69. e this idea when simplifying rational expressions 4 x 2 x 2 x xi x 6 x 3 x 2 x 2 2 x z i x 2 2 Xcb 3 o x 3 Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 50 Chapter O Some Basic Concepts of Algebra A Review Multiplying and Dividing Rational Expressions Multiplication of rational expressions is based on the following property BE Lu b d bd In other words we multiply numerators and we multiply denominators and express the final product in simplified form Study the following examples carefully and pay special attention to the formats used to organize the computational work 2 ax 8y 3 8 x y 2y 4y 9x 49 3 2 8 x 12x y 24xy 12 24 y 2x 12x y 12x y 24xy 24xy an 18xy 56y 18 56 x y Ty 18xy 18xy 56y 56y 3 7 y so the product is positive y x 2 Z x4 2 L 1 x 4 y y x 2 x 2 y x 2 y Lok ator X x T x4 T x 4 x 4 xt5 x xx x 5G 1 0 1 x x 5 To divide rational expressions we merely apply the following property a c ad ad b d bc bc That is the quotient of two rational expressions is the product of the first expres sion times the reciprocal of the second Consider the following examples 16x y 9xy 1600 amp y 16 8 16x 24xy gt 8x y i 24xy Oxy i 24 9 32 y 277 3 y
70. ed or duplicated in whole or in part Licensed to iChapters User EXAMPLE 7 0 4 Factoring Polynomials 43 By checking the middle terms we find that x 4 5x 2 yields the desired middle term of 18x Thus 5x 18x 8 x 4 5x 2 El Factor 4x 6x 9 Solution The first term 4x and the positive signs of the middle and last terms indicate that the binomials are of the form x __ 4x __ or 2x __ 2x __ Because the factors of the last term 9 are 1 and 9 or 3 and 3 we have the follow ing possibilities to try x 1 4x 9 x 9 4x 1 x 3 4x 3 2x 1 2x 9 2x 3 2x 3 None of these possibilities yields a middle term of 6x Therefore 4x 6x 9 is not factorable using integers Certainly as the number of possibilities increases this trial and error tech nique for factoring becomes more tedious The key idea is to organize your work so that all possibilities are considered We have suggested one possible format in the previous examples However as you practice such problems you may devise a format that works better for you Whatever works best for you is the right approach There is another more systematic technique that you may wish to use with some trinomials It is an extension of the technique we used earlier with trinomials where the coefficient of the squared term was one To see the basis of this tech nique consider the follow
71. ed by using either because by property 3 above they are the same quantity Cartesian Coordinate System Just as real numbers can be associated with points on a line pairs of real numbers can be associated with points in a plane To do this we set up two number lines one Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 8 Chapter O Some Basic Concepts of Algebra A Review vertical and one horizontal perpendicular to each other at the point associated with zero on both lines as shown in Figure 0 8 We refer to these number lines as the horizontal axis and the vertical axis or together as the coordinate axes They partition a plane into four regions called quadrants The quadrants are numbered counterclockwise from I through IV as indicated in Figure 0 8 The point of inter section of the two axes is called the origin 5 4 II 3 I 2 1 0 gt zaa 12345 2 III 3 IV E ES Figure 0 8 The positive direction on the horizontal axis is to the right and the positive direction on the vertical axis is up It is now possible to set up a one to one corre spondence between ordered pairs of real numbers and the points in a plane To each ordered pair of real numbers there corresponds a unique point in the plane and to each point in the plane there corresponds a unique ordered pair of real num bers A part of this corresponde
72. eet the following type of simplification problem in calculus x 1 x x 1 x 1 E 1 x x E E d x 1y x 1 x 1 x x 1 P 7 x 1 0 7 Relationship Between Exponents and Roots 75 For Problems 49 56 rationalize the denominator and express the final answer in simplest radical form 5 s 2 49 52 S5 N A 54 55 56 AW x Vab 57 Simplify each of the following expressing the final result as one radical For example Wh 31212 31 4 43 e VV TA X b d 59 Explain how you would evaluate 2777 without a calculator x13 x 12 x 1 1 1 1 x 1 x 1 For Problems 60 65 simplify each expression as we did in the previous example 2 x 1 x x 1 0 2 x Dp 60 Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 76 Chapter 0 Some Basic Concepts of Algebra A Review 2 2x 1 2x 2x 1 0 2 3x 5 x 3x 0 9 Ps Ox PP M Gui 2x 4x 1 2x Ax 1 9 2 3 2x 2x 2x 09 4x 192 Qxy5p x 4 2xy x x 1 x 4 x 2x P E Graphing calculator activities 66 Use your calculator to evaluate each of the following 2x 02 63 68 Use your calculator to evaluate each ofthe following a V1728 b V 58
73. f Property 0 1 can be justified by using Definition 0 2 For example to justify part 1 we can reason as follows b b bbb b bbb b n factors m factors of b of b Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User EXAMPLE 1 mem 5 EXAMPLE 3 EXAMPLE 4 0 2 Exponents 19 bbb b n m factors of b pun Similar reasoning can be used to verify the other parts of Property 0 1 The fol lowing examples illustrate the use of Property 0 1 along with the commutative and associative properties of the real numbers The steps enclosed in the dashed boxes can be performed mentally fe SSS mmo n 3xy 4x y 913 4 x xo y y ee ee D Bets 12x mH 2y P CDG by anb 32y py pm m aM y 2 a Bs Dy b b d E D DER pm 56x r z n Y iI 8X4 b L prom whenn gt m 7x Il b 8x gt oO Zero and Negative Integers As Exponents Now we can extend the concept of an exponent to include the use of zero and nega tive integers First let s consider the use of zero as an exponent We want to use zero in a way that Property 0 1 will continue to hold For example if b b p is to hold then x x should equal x which equals x In other words x acts like 1 because x x x Look at the following definition DEFINITION 0 3 If b is a
74. g polynomials 1 a bY a 2ab b 2 a by a 2ab b 3 a b a b a b 4 a by a 3a b 3ab b 5 a b a 3a b 300 b Chapter O Summary 85 SUMMARY Be sure you know how to do the following 1 Factor out the highest common monomial factor 2 Factor by grouping 3 Factor a trinomial into the product of two binomials 4 Recognize some basic factoring patterns a 2ab b a by a 2ab b a by a b 2 a b a b a b a b a ab b a gt b a b a ab b Be sure that you can simplify add subtract multiply and divide rational expressions using the following properties and definitions Gols oi P pin ua a a pe DES b a C ac SU TOM Ala a d 40 b d bc be 6 idi CG G iG e uu C C C Be sure that you can simplify add subtract multiply and divide radicals using the following definitions and properties 1 Vb a ifandonlyif a b 2 Vie DWG EE Bram c The following definition provides the link between ex ponents and roots prn Zy p 6 Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 86 Chapter O Some Basic Concepts of Algebra A Review This link along with the properties of exponents allows us 1 to multiply and divide some r
75. he real number line indicates a one to one correspondence between the set of real numbers and the points on a line In other words to each real number there corre sponds one and only one point on the line and to each point on the line there cor responds one and only one real number The number that corresponds to a partic ular point on the line is called the coordinate of that point 1 1 mq ccm 2 2712 T e _ ee 9 9 09 9 9 9 9 9 9 ee e e gt 5 4 3 2 1 0 1 2 3 4 5 Figure 0 2 Many operations relations properties and concepts pertaining to real numbers can be given a geometric interpretation on the number line For example Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 6 Chapter O Some Basic Concepts of Algebra A Review the addition problem 1 2 can be interpreted on the number line as shown in Figure 0 3 EL D C2 2 3 2 Figure 0 3 b a c d The inequality relations also have a geometric interpretation The statement Ee A 9 a gt b read a is greater than b means that a is to the right of b and the statement Figure 0 4 c lt d read c is less than d means that c is to the left of d see Figure 0 4 The property x x can be pictured on the number line in a sequence of steps See Figure 0 5 1 Choose a point that has a coor
76. iChapters User 28 Chapter O Some Basic Concepts of Algebra A Review 117 Use your calculator to estimate each of the follow 118 Use your calculator to estimate each of the follow ing Express final answers in scientific notation ing Express final answers in ordinary notation with the number between 1 and 10 rounded to the rounded to the nearest one thousandth nearest one thousandth a 1 09 b 1 08 4 10 a 4576 b 719 e 1 147 d 1 12 12 6 c 28 d 8619 e 0 785 f 0 492 e 314p f 145 723 0 3 Polynomials Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Recall that algebraic expressions such as 5x 6y 2x ly 14a b 5x and 17ab c are called terms Terms that contain variables with only nonnegative in tegers as exponents are called monomials Of the previously listed terms 5x 6y 14a b and 17ab c are monomials The degree of a monomial is the sum of the exponents of the literal factors For example 7xy is of degree 2 whereas 14a b is of degree 3 and 17ab c is of degree 6 If the monomial contains only one vari able then the exponent of that variable is the degree of the monomial For ex ample 5x is of degree 3 and 8y is of degree 4 Any nonzero constant term such as 8 is of degree zero A polynomial is a monomial or a finite sum of monomials Thus all of the fol lowing are examples of po
77. iction is needed From Definition 0 5 we see that if n is any positive integer greater than 1 and Wb exists then Vby b For example V4 4 N 8y 8 and W 81 81 Furthermore if b 0 and n is any positive integer greater than 1 or if b lt 0 and n is an odd positive inte ger greater than 1 then Wb b For example Ve 4 V 2y 2 and 6 6 But we must be careful because V 2 2 2 and W 2 2 Simplest Radical Form Let s use some examples to motivate another useful property of radicals V16 25 V400 20 and V16 V25 4 5 20 W8 27 V216 6 and V8 V27 22 3 6 V 8 64 V 512 8 and V 8 V 64 2 4 8 In general the following property can be stated PROPERTY 0 3 V bc Vb We if Wb and Wc are real numbers Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 0 6 Radicals 63 Property 0 3 states that the nth root of a product is equal to the product of the nth roots The definition of nth root along with Property 0 3 provides the basis for changing radicals to simplest radical form The concept of simplest radical form takes on additional meaning as we encounter more complicated expressions but for now it simply means that the radicand does not contain any perfect powers of the index Consider the following examples of reductions to simplest radical form V45 V9
78. imately 5 900 000 000 000 miles and this can be written as 5 9 10 The weight of an oxygen molecule is approximately 0 000000000000000000000053 of a gram and this can be expressed as 5 3 10 To change from ordinary decimal notation to scientific notation the follow ing procedure can be used Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 24 Chapter O Some Basic Concepts of Algebra A Review Thus we can write 0 00092 9 2 10 872 000 000 8 72 10 5 5 1217 5 1217 10 To change from scientific notation to ordinary decimal notation the following pro cedure can be used Thus we can write 3 14 10 31 400 000 7 8 10 0 0000078 Scientific notation can be used to simplify numerical calculations We merely change the numbers to scientific notation and use the appropriate properties of ex ponents Consider the following examples Perform the indicated operations 0 00063 960 000 V 90 00 a 3200y0 0000021 V 90 000 Solution Tm 3 0 00063 960 000 6 3 10 9 6 10 a 320000000021 3 210 2 1 10 6 3 9 6 10 3 2 2 1 105 9 10 90 000 Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 0 2 Exponents 25 b V 90 000
79. ing a binomial as the next two examples illustrate 3x 2 3x 3 3xy 2 3 3x Q2 2 27x 54x 36x 8 Sx 2y Sx 3 5xY Qy 3 5x QyY 2 125x 150x y 60xy 8y Keep in mind that these multiplying patterns are useful shortcuts but if you forget them simply revert to applying the distributive property Binomial Expansion Pattern It is possible to write the expansion of a b where n is any positive integer without showing all of the intermediate steps of multiplying and combining similar terms To do this let s observe some patterns in the following examples each one can be verified by direct multiplication a b a b a by a 2ab b a b a 3a b 3ab b a b Jat 4200 6a b 4dab gt b a b a 5a b 10a b 10a b 5ab b First note the patterns of the exponents for a and b on a term by term basis The exponents of a begin with the exponent of the binomial and decrease by 1 term by term until the last term which has a 1 The exponents of b begin with zero b 1 and increase by 1 term by term until the last term which contains b to the power of the original binomial In other words the variables in the expansion of a b have the pattern a a b a p TN ab b where for each term the sum of the exponents of a and b is n Next let s arrange the coefficients in a triangular formation this yields
80. ing general product px r qx s px qx px s r qx r s pq x ps x rq x rs pq x ps rq x rs Notice that the product of the coefficient of x and the constant term is pqrs Like wise the product of the two coefficients of x ps and rq is also pqrs Therefore the coefficient of x must be a sum of the form ps rq such that the product of the co efficient of x and the constant term is pqrs Now let s see how this works in some specific examples Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 44 Chapter O Some Basic Concepts of Algebra A Review EXAMPLE 8 EXAMPLE 9 Factor 6x 17x 5 Solution umi 6x 17x 5 Sum of 17 Product of 6 5 30 We need two integers whose sum is 17 and whose product is 30 The integers 2 and 15 satisfy these conditions Therefore the middle term 17x of the given trinomial can be expressed as 2x 15x and we can proceed as follows 6x 17x 5 6x 2x 15x 5 2x 3x 1 5 3x 1 3x 1 Qx 5 Ej Factor 5x 18x 8 Solution eR ILI 5x 18x 8 Sum of 18 Product of 5 8 40 We need two integers whose sum is 18 and whose product is 40 The integers 20 and 2 satisfy these conditions Therefore the middle term 18x of the trino mial can be written 20x 2x and we can fa
81. inverse is the identity element for addition For example 16 and 16 are additive inverses and their sum is zero The additive inverse of zero is zero The real number 1 a is called the multiplicative inverse or reciprocal of a The product of a number and its multiplicative inverse is the identity element for 1 1 1 multiplication For example the reciprocal of 2 is y and 2 3 32 1 The product of any real number and zero is zero For example 17 0 0 17 0 The product of any real number and 1 is the opposite of the real number For example 1 52 52 1 52 The distributive property ties together the operations of addition and mul tiplication We say that multiplication distributes over addition For example 7 3 8 7 3 7 8 Furthermore because b c b c it follows that multiplication also distributes over subtraction This can be expressed symbolically as a b c ab ac For example 6 8 10 6 8 6 10 Graphing Utilities The term graphing utility is used in current literature to refer to either a graphing calculator see Figure 0 12 or a computer with a graphing software package We will frequently use the phrase use a graphing calculator to mean either a graph ing calculator or a computer with an appropriate software package We will intro duce various features of graphing calculators as we need them in the text Because so many different types of g
82. is merely an other multiplication problem 3x 4y 3x 4y 3x 4y 9x 24xy 16y Hint When squaring a binomial be careful not to forget the middle term That is x 5 x 25 instead x 5 x 10x 25 x 4 x 4 x 4 x 4 x 4 x 8x 16 x x 8x 16 4 x 8x 16 x 8x 16x 4x 32x 64 x 12x 48x 64 Special Patterns In multiplying binomials you should learn to recognize some special patterns These patterns can be used to find products and some of them will be helpful later when you are factoring polynomials The three following examples illustrate the first three patterns respectively 2x 3 2x 2 2x 3 GY 4x 12x 9 5x 2 xY 2Gx Q 2 25x 20x 4 3x 2y 3x 2y Gx 27 9x 4y In the first two examples the resulting trinomial is called a perfect square trino mial it is the result of squaring a binomial In the third example the resulting bi nomial is called the difference of two squares Later we will use both of these pat terns extensively when factoring polynomials Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 0 3 Polynomials 33 The cubing of a binomial patterns are helpful primarily when you are multi plying These patterns can shorten the work of cub
83. ive integers 18 The real numbers For Problems 19 32 use the following set designations N x x is a natural number W x x is a whole number I x x is an integer Q x x is a rational number H x x is an irrational number R x x is a real number Place C or in each blank to make a true statement 19 N R 20 R N 21 N I 22 I Q Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 16 Chapter O Some Basic Concepts of Algebra A Review 23 H Q 24 Q H 25 W 1 26 N W 27 7 w 28 I N 29 0 2 4 W 30 1 3 5 7 I 31 2 1 0 1 2 W 32 0 3 6 9 N For Problems 33 42 list the elements of each set For example the elements of x x is a natural number less than 4 can be listed 1 2 3 33 x x is a natural number less than 2 34 x x is a natural number greater than 5 35 n n is a whole number less than 4 36 y y is an integer greater than 3 37 y y is an integer less than 2 38 n n is a positive integer greater than 4 39 x x is a whole number less than 0 40 x x is a negative integer greater than 5 41 n n is a nonnegative integer less than 3 42 n n is a nonpositive integer greater than 1 43 Find the distance on the real number line between two points whose coordinates are the following a 17 and 35 b 14 and 12 c 18 and
84. l factor is called the numerical coefficient Thus in 8xy the x and y are literal factors and 8 is the numerical coefficient Because 1 z z the numerical coefficient of the term z is understood to be 1 Terms that have the same literal factors are called Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User EXAMPLE 1 p 25 0 1 Some Basic Ideas 13 similar terms or like terms The distributive property in the form ba ca b c a provides the basis for simplifying algebraic expressions by combining similar terms as illustrated in the following examples 3x 5x 3 5 x 6xy 4xy 6 4 xy 4x x 4x 1x 8x 2xy 4 1 Xx 3x Sometimes we can simplify an algebraic expression by applying the distribu tive property to remove parentheses and combine similar terms as the next ex amples illustrate 4 x 2 3 x 6 4 x AQ 3 x 3 6 4x 8 3x 18 7x 26 5 y 3 2 7 8 509 230 2 y 2 8 5y 15 2y 16 Ty 1 An algebraic expression takes on a numerical value whenever each variable in the expression is replaced by a real number For example when x is replaced by 5 and y by 9 the algebraic expression x y becomes the numerical expression 5 9 which is equal to 14 We say that x y has a value of 14 when x 5 and y 9 Consider the following
85. l s triangle 1 4 6 4 1 to obtain the coefficients a b at 4a b 6a by 4a bp by a 4a b 6a b 4ab b B 1 Solution p BF 7 _ Let 2x a and 3y b The coefficients 1 5 10 10 5 1 come from the fifth row of Pascal s triangle 2x 3y 2x 5 2x 3y 102x 3y 102x F 3y P 5 2x 3y By 32x5 240x y 720x y 1080x y 810xy 243y B Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 0 3 Polynomials 35 Dividing Polynomials by Monomials In Section 0 5 we will review the addition and subtraction of rational expressions using the properties LET PEE ME d a c a c b b b US b b These properties can also be viewed as Qe a c a C c and LC ES b b b b b b Together with our knowledge of dividing monomials these properties provide the basis for dividing polynomials by monomials Consider the following examples 18x 24x 18x 24x 3x 4 6x 6x 6x d ae 35x 55221 35x y 0 55221 4 5 ave Txy ix 5xy 5xy 5xy Therefore to divide a polynomial by a monomial we divide each term of the poly nomial by the monomial As with many skills once you feel comfortable with the process you may then choose to perform some of the steps mentally Your work could take the following format 40x y 72z5y
86. licated in whole or in part Licensed to iChapters User 0 8 Complex Numbers 81 Conjugates are used to simplify an expression such as 3i 5 2i which indi cates the quotient of two complex numbers To eliminate i in the denominator and to change the indicated quotient to the standard form of a complex number we can multiply both the numerator and denominator by the conjugate of the denominator 3i 3i 5 2i 542i 542i 5 2i 3i 5 2i 5 2i 5 2i _ 15i 62 25 47 158 6 1 25 4 1 6 15i 29 26 15 29 99 The following examples further illustrate the process of dividing complex numbers 2 3i 2 31 4 7i 4 T7i 4 71 4 7 2 3i 4 7i 4 7i 4 7i 8 147 127 217 16 49 8 2i 21 1 16 49 1 29 2i 29 2 i 65 65 65 4 5i 4 5i 2i 2i 2i 2i 4 5Si 2i 2i 2i 8i 10 4P 8i 10 1 4 1 10 8i 5 a l 3ji 4 2 7 Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 82 Chapter O Some Basic Concepts of Algebra A Review For a problem such as the last one in which the denominator is a pure imaginary number we can change to standard form by choosing a multiplier other than the conjugate of the denominator Consider the following alternative approach 2i 2i 4 5i
87. lynomials 4x 3x 2x 4 7x 6x 5x 2x 1 1 2 3x y 2 a Zb 14 xX y y 51 3 In addition to calling a polynomial with one term a monomial we classify poly nomials with two terms as binomials and those with three terms as trinomials The degree of a polynomial is the degree of the term with the highest degree in the poly nomial The following examples illustrate some of this terminology The polynomial 4x y is a monomial in two variables of degree 7 The polynomial 4x y 2xy is a binomial in two variables of degree 3 The polynomial 9x 7x 1 is a trinomial in one variable of degree 2 Addition and Subtraction of Polynomials Both adding polynomials and subtracting them rely on the same basic ideas The commutative associative and distributive properties provide the basis for re arranging regrouping and combining similar terms Consider the following addi tion problems Licensed to iChapters User 0 3 Polynomials 29 Ax 5x 1 7x 9x 4 Ax 7x 5x 9x 1 4 11x 4x 5 5x 3 3x 2 8x 6 5x 3x 8x 3 2 6 16x45 The definition of subtraction as adding the opposite a b a b ex tends to polynomials in general The opposite of a polynomial can be formed by taking the opposite of each term For example the opposite of 3x 7x 1 is 3x 7x 1 Symbolically this is expressed as 3x 7x 1
88. mber can be traced down through the tree Here are some examples 7 is real rational an integer and positive 2 i 73 is real rational a noninteger and negative V 7 is real irrational and positive 0 59 is real rational a noninteger and positive Licensed to iChapters User 0 1 Some Basic Ideas 5 The concept of a subset is convenient to use at this time A set A is a subset of another set B if and only if every element of A is also an element of B For ex ample if A 1 2 and B 1 2 3 then A is a subset of B This is written A C B and is read A is a subset of B The slash mark can also be used here to denote nega tion If A 1 2 4 6 and B 2 3 7 we can say A is not a subset of B by writing A B The following statements use the subset vocabulary and symbolism they are represented in Figure 0 1 Reals Rationals Integers Whole numbers Figure 0 1 1 The set of whole numbers is a subset of the set of integers 071 2 3 1 feng 72 71 0 12 2 The set of integers is a subset of the set of rational numbers 2 1 0 1 2 C x x is a rational number 3 The set of rational numbers is a subset of the set of real numbers x x is a rational number C y y is a real number Real Number Line and Absolute Value It is often helpful to have a geometric representation of the set of real numbers in front of us as indicated in Figure 0 2 Such a representation called t
89. metimes the result 1 of applying the definition b p Our final example illustrates this idea Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 0 5 Rational Expressions 22x e y Simplify 95 x oy Solution b First let s apply b 2 L4 X lt 2x7 y x 3y x uo Now we can proceed as in the previous examples 2 1 2 1 ha 2 hie 2 x y 25 QUY y 3 xy 3 tog a 205 y y 7 2y xy i xly 3x PROBLEM SET 0 5 For Problems 1 18 simplify each rational expression 1 11 13 14 14x y 21xy 63xy 817 2607 3xyY a c 7a 12 6a 27 2x 3x 14x oxiy 7xy 18y p y x xy 2y 2y 2xy xy y 2 10 16x y 24x y 16xy 24x y 12xy 12y 8x 4xy 2x y 57 2x3 2y i e 16 26xy 4x 437 x y 65y 9 27x 8y TE ER L y 3x 15x 2xy 10y xX xy x 64 a 3036 3x 1lx 4 6a b 6x x 15 ge 10x 3 simplest form 3x x 0 20 xi 9 Sy 24x y ax 3x 2ay 6y 2 14xy 24x y 22 2ax 6x ay 3y 18y 3570 Ta b 3a 23 zx 24 9ab 2qp 5 36 6 dE 26 X 6 x 6x 2x 6x 2xy 6y For Problems 19 68 perform the indicated operations involvi
90. ms 9 44 express each in simplest radical form All variables represent positive real numbers 1 V81 2 V49 i V24 10 V 54 el 9 3 V 125 4 N 81 T 11 V112 12 628 5 6 49 64 13 3V 44 14 5V 68 27 64 3 3 an E 8 4 z 15 20 16 avn Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 17 V12x 18 V45xy 19 V 6Ax y 20 3 324 21 T VAI 22 32 23 W 128 24 N 54x 25 V 16x 26 W 81x y 27 W 48x 28 V 162x5y 12 175 29 55 30 37 7 V35 31 32 8 V7 V5 V 18 as 03 NES 7V6 V 2y 37 5 38 V 12x 18x SM Cie bA 50302 3 3 41 W217 42 gt V A 2x 4 2y 44 Vv 12xy ee 3x E 3x2y For Problems 45 52 use the distributive property to help simplify each For example 3V8 5V2 3V4V2 5V2 6V2 5V2 6 4 5 V2 11V2 45 5V 12 2V3 46 4 50 932 47 2V28 3V 63 8V7 0 6 Radicals 69 48 40 2 21 16 W 54 5 3 2 1 49 c V48 V2 50 Sva c V90 2V8 3V18 v50 1 d 5 2 3V s4 5N 16 SET p For Problems 53 68 multiply and express the results in simplest radical form All variables represent nonnega tive real numbers 53 4 V 3 6V 8 54 5V 8 3 V7 55 2V 3 5V2 4V 10 56 3V6 2 V8 312 57 3Vx V 6xy V 8y 58 Voy V8x V10y 59 V3 2 V3 5 60 V2 3 V2 4
91. multiple of the de nominators of those numbers is the least common denominator LCD This con cept of a least common denominator can be extended to include polynomials RD 3x41 Add 3 Solution By inspection we see that the LCD is 12 HPC CSIC 3 x 2 4 3x 1 12 12 3x 6 12x 4 12 15x 10 pe eee a a Bl 12 Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 52 Chapter O Some Basic Concepts of Algebra A Review EXAMPLE 2 Perform the indicated operations x43 2x l Eee 10 15 18 Solution If you cannot determine the LCD by inspection then use the prime factored forms of the denominators 10 2 5 15 3 5 18 2 3 3 The LCD must contain one factor of 2 two factors of 3 and one factor of 5 Thus the LCD is2 3 3 5 90 x 3 26 uA 5 6 10 15 18 10 9 15 6 18 3 _ X x 3 62x c1 5 x 2 90 90 90 9x 27 12x 6 5x 10 7 90 _ 16x 43 90 The presence of variables in the denominators does not create any serious difficulty our approach remains the same Study the following examples very care fully For each problem we use the same basic procedure 1 Find the LCD 2 Change each fraction to an equivalent fraction having the LCD as its denomi nator 3 Add or subtract numerators and place this result over the LCD 4 Look for pos
92. n can be written xVx Such simplification can also be done in exponential form as follows 3 3 N x x43 38 IB y x18 We Note the use of this type of simplification in the following examples Perform the indicated operations and express the answers in simplest radical form a VW b V2N A c UA Solutions a NV _ PR gt 5634 le E g x VO b vV2N A 21 2 4173 21 2 22 43 21 2 22 5 21 242 3 97 6 26 6 91 6 2x2 3 2 Ver oF quy am 33 2 1 3 376 N 3 31 3 31 3 31 3 395 31 6 313 a The process of rationalizing the denominator can sometimes be handled more easily in exponential form Consider the following examples which illustrate this procedure Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 74 EXAMPLE 2 Chapter O Some Basic Concepts of Algebra A Review ems Rationalize the denominator and express the answer in simplest radical form a b Wx Vy Solutions 2 2 2 x 295 x a Wa x yh wm uw Wy MM y xi y x6 y6 6 vy b um yum x Vy y yiP y y y y xy V Note in part b that if we had changed back to radical form at the step y we would have obtained the product of two radicals Vx Vy in the numerator In stead we used the exponential form to find this product and express the final result with a single radical in the n
93. n whole or in part Licensed to iChapters User A B 3 3 3 2 1 0 1 2 3 4 Figure 0 7 0 1 Some Basic Ideas 7 Symbolically absolute value is denoted with vertical bars Thus we write 2 2 3 3 and 0 0 More formally the concept of absolute value is defined as follows DEFINITION 0 1 For all real numbers a 1 Ifa 0 then a a 2 Ifa lt 0 then a a According to Definition 0 1 we obtain 6 6 by applying part 1 0 0 by applying part 1 7 7 7 by applying part 2 Notice that the absolute value of a positive number is the number itself but the absolute value of a negative number is its opposite Thus the absolute value of any number except zero is positive and the absolute value of zero is zero Together these facts indicate that the absolute value of any real number is equal to the ab solute value of its opposite All of these ideas are summarized in the following properties Properties of Absolute Value The variables a and b represent any real number 1 a 2 0 2 a a 3 a b b a a bandb a are opposites of each other In Figure 0 7 the points A and B are located at 2 and 4 respectively The distance between A and B is 6 units and can be calculated by using either 2 4 or 4 2 In general if two points on a number line have coordinates x and x then the distance between the two points is determin
94. nce is illustrated in Figure 0 9 For example the or dered pair 3 2 means that the point A is located 3 units to the right of and 2 units up from the origin Likewise the ordered pair 3 5 means that the point D is located 3 units to the left of and 5 units down from the origin The ordered pair 0 0 is associated with the origin O R24 AG 2 CC4 0 100 0 E 5 2 D 3 5 e Figure 0 9 Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 0 1 Some Basic Ideas 9 In general we refer to the real numbers a and b in an ordered pair a b as sociated with a point as the coordinates of the point The first number a called the abscissa is the directed distance of the point from the vertical axis measured par allel to the horizontal axis The second number b called the ordinate is the di rected distance of the point from the horizontal axis measured parallel to the ver tical axis Figure 0 10 Thus in the first quadrant all points have a positive abscissa and a positive ordinate In the second quadrant all points have a negative abscissa and a positive ordinate We have indicated the sign situations for all four quadrants in Figure 0 11 This system of associating points in a plane with pairs of real numbers is called the rectangular coordinate system or the Cartesian coordinate system A A a
95. nd Difference of Two Cubes Earlier in this section we discussed the difference of squares factoring pattern We pointed out that no analogous sum of squares pattern exists that is a polynomial such as x 9 is not factorable using integers However there do exist patterns for both the sum and the difference of two cubes These patterns come from the fol lowing special products x ye ay y x x xy y y x xy ys x x y xy x y xy y x y x y G xy y x x xy y y x xy y x54 x y H xy x y xy y x3 y3 Thus we can state the following factoring patterns x y x y x xy y x y x y x xy y Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 46 Chapter O Some Basic Concepts of Algebra A Review Note how these patterns are used in the next three examples x48 23 x 2 x 2x 4 8x3 27y xy 37 2x 3y 4x Oxy 9y 8a 125b 2a7 5b 2a 5b 4a 10a b 25b We do want to leave you with one final word of caution Be sure to factor completely Sometimes more than one technique needs to be applied or perhaps the same technique can be applied more than once Study the following examples very carefully 2x 8 2 x 4 2 x 2 x 2 3x 18x 24 3
96. ng rational expressions Express final answers in 5xy 18x y 8y 15 6xy 30x y 9y 48x 9a c 2100 12b 14 2 6 C a a 8a 4 Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 58 27 28 29 30 31 32 33 34 37 38 39 41 43 45 47 49 51 Chapter O Some Basic Concepts of Algebra A Review 5a 20a amp a 12 2 3 7 3 2023 gt P 83 Sh 2 4 3n 6 a 2a a 16 n n tt 81 du 117 21 55 3 5 3 7 67 9 54 87 21 yti x 1 x 1 x 5xy 6y 2x 15xy 18y 56 5 5x 30 x xy y xy 4y x x 6x x 6 10n 21n 10 2r 6n 56 57 5 4 5n 3377 14 2m 3n 20 x 4 10x 21 x 12x 27 9y 127 58 8 10 x 12x 36 x 6x d 30 18 a4 70 30 x 4xy Ay Ax 3xy 10y 59 5 2 7xy 20x y 25xy 21 x 6x 16 2x 3x Pie ae a 14x 4 21 60 4 7 2x3 10x 3x3 27x x 6x 27 Pl wf tee 17 a 4ab 4b 3a 5ab 2b a 4b 61 3x 2 6a 4ab 60 00 b 8a 4b x 6x 9 x 3 x 4 2x 1 377 1 n 2 6 9 ep i 62 m 6 E 4 m 9 12 9 x 6x 49 xl ye ox x 1 63 ay 6 8 pre nos pm 2 x 2 k e Fo aS x i 5 6 15 x 7 x 49 2n 1
97. not be copied scanned or duplicated in whole or in part Licensed to iChapters User 68 Chapter O Some Basic Concepts of Algebra A Review Thus if x 0 then Vx x but if x lt 0 then V x Using the concept of ab solute value we can state that for all real numbers V x x Now consider the radical V x Because x is negative when x is negative we need to restrict x to the nonnegative real numbers when working with V x Thus we can write ifx 0 then Vi Vx2Vx x Vx and no absolute value sign is needed Finally let s consider the radical Vs Let x 2 then Vx W 23 N 8 2 Let x 2 then Wx3 W 2 W 8 2 Thus it is correct to write 3 x x for all real numbers and again no absolute value sign is needed The previous discussion indicates that technically every radical expression with variables in the radicand needs to be analyzed individually to determine the necessary restrictions on the variables However to avoid having to do this on a problem by problem basis we shall merely assume that all variables represent pos itive real numbers Let s conclude this section by simplifying some radical expressions that con tain variables 72137 V36x1y 5 V2xy 6xy V 2xy V 403 y Mm N 8x3y5 N sxy Dey V Suy s V5 V5 V3a Visa Visa V 12 Via V3a V36 6a 3 3 2E SUE 32x Wax Wax N2x W8x3 2x PROBLEM SET 0 6 For Problems 1 8 evaluate For Proble
98. not be copied scanned or duplicated in whole or in part Licensed to iChapters User ES Thoughts into words 83 84 Describe how to multiply two binomials Describe how to multiply a binomial and a trinomial 0 4 Factoring Polynomials 37 85 Determine the number of terms in the product of x y and a b c d without doing the mul tiplication Explain how you arrived at your answer a Graphing calculator activities 86 87 88 89 Use the computing feature of your graphing calcu lator to check at least one real number for your an swers for Problems 29 40 Use the graphing feature of your graphing calcula tor to give visual support for your answers for Prob lems 47 52 Some of the product patterns can be used to do arith metic computations mentally For example let s use the pattern a b a 2ab b to compute 31 mentally Your thought process should be 31 30 1 30 2 30 1 1 961 Compute each of the following numbers mentally and then check your answers with your calculator a 21 b 41 c 712 d 32 e 52 f 82 Use the pattern a b a 200 b to com pute each of the following numbers mentally and then check your answers with your calculator a 19 b 29 c 49 d 79 e 38 f 58 0 4 Factoring Polynomials 90 Every whole number with a units digit of 5 can be represented by the expression 10x 5 where x is a whol
99. on of numerical and alge braic expressions involving sums and differences In such cases Definition 0 4 can be used to change from negative to positive exponents so that we can proceed in the usual ways Solution 2 37 Me ll N oo ge i y w oo z 2 24 Simpify 47 3 2 Solution E Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 0 2 Exponents 23 Figure 0 15 shows calculator windows for Examples 7 and 8 Note that the an swers are given in decimal form If your calculator also handles common fractions then the display window for Example 7 may appear as in Figure 0 16 Figure 0 15 Figure 0 16 EXAMPLE 9 Express a b as a single fraction involving positive exponents only Solution 1 1 CO Gs GSC aJNB NP Na EE 7j ab ab 22 0 7 ab Scientific Notation The expression 1 10 where n is a number greater than or equal to 1 and less than 10 written in decimal form and k is any integer is commonly called scientific notation or the scientific form of a number The following are examples of numbers expressed in scientific form 4 23 10 _ 8176 00 5 0200 1 10 Very large and very small numbers can be conveniently expressed in scien tific notation For example a light year the distance that a ray of light travels in one year is approx
100. ore to subtract c di from a bi we add the additive inverse of c di a bi c di a bi c di a c b d i The following examples illustrate the subtraction of complex numbers 9 87 5 37 9 5 8 3 7 4 57 3 27 4 107 3 4 2 10 i 1 8i 141 244 ES amp 1 2 3 4 27 3 4 A3 20 4 6 Multiplying and Dividing Complex Numbers Because i 1 the number i is a square root of 1 so we write i V 1 It should also be evident that i is a square root of 1 because 2 2 20 2 1 Therefore in the set of complex numbers 1 has two square roots namely i and i This is expressed symbolically as i V 1 and i V l1 Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 0 8 Complex Numbers 79 Let s extend the definition so that in the set of complex numbers every negative real number has two square roots For any positive real number b i Vby i b 1 b b Therefore let s denote the principal square root of b by V b and define it to be V b iVb where b is any positive real number In other words the principal square root of any negative real number can be represented as the product of a real number and the imaginary unit 7 Consider the following examples V 4 i
101. pe 65 7 4 16a b 20b 240 32b m 16 m 4 n 2 E15 8 RENE 66 ty etn 1 m 4n 6 m 5n 3 mc 1 n i n 1 3 2 a ae 67 2x 1 3x 2 x 37 6x 4y x 3x 4 x7 3x 28 3 2 5 3 3x 4 2x 1 68 2x41 wed M 2x 3 2x 9x 5 3x 11x 20 4 3 6 3 8 5 z 48 69 Consider the addition problem 4 x 7x x x 8x x x 2 2X TNT 3 fui 5 Note that the denominators are opposites of each 4 a 2 m 1 a l other If the property ES E is applied to the 3 3x 2 2x aeq o c 52 mie Sema second fraction we obtain CET eT Thus Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User we can proceed as follows 8 E 5 _ 8 5 2 2x x 2 XxX 2 8 5 3 X Mo aga Use this approach to do the following problems 7 2 x 1 1 x b 2 H 2 22 1 T 2x 4 1 0 3 3 a d 10 5 a 9 9 a x _ 2x 3 x 1 l t x 3x 28 x 4 4 x For Problems 70 92 simplify each complex fraction 217 A Z7 P p A X 70 71 3 10 hc 2 x y y y 1 1 cup 1 des 72 73 bs 3 4 1 1 y X 2 2 4 n 1 74 P 75 1 5 1 n 4 n 1 H5 Thoughts into words 93 What role does factoring play in the simplifying of rational expressions 76 78 80 82 84 86 87 88 89 94 0
102. r Be sure of the following key concepts from this chapter set null set equal sets subset natural numbers whole numbers integers rational numbers irrational num bers real numbers complex numbers absolute value similar terms exponent monomial binomial polyno mial degree of a polynomial perfect square trinomial factoring polynomials rational expression least com mon denominator radical simplest radical form root and conjugate of a complex number The following properties of the real numbers provide a basis for arithmetic and algebraic computation closure for addition and multiplication commutativity for addi tion and multiplication associativity for addition and multiplication identity properties for addition and mul tiplication inverse properties for addition and multipli cation multiplication property of zero multiplication property of negative one and distributive property The following properties of absolute value are useful 1 a 2 0 2 a a 3 la b b a a and b are real numbers The following properties of exponents provide the basis for much of our computational work with polynomials 1 b bm pum 2 b n b mn 3 ab a b We ae 4 tee 2 b b 5 b b m and n are rational numbers and a and b are real numbers except b 0 whenever it appears in the denominator The following product patterns are helpful to recognize when multiplyin
103. r then b 066 b n factors of b Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 18 Chapter O Some Basic Concepts of Algebra A Review The number b is referred to as the base and n is called the exponent The expres sion b can be read b to the nth power The terms squared and cubed are commonly associated with exponents of 2 and 3 respectively For example b is read b squared and b as b cubed An exponent of 1 is usually not written so b is simply written b The following examples illustrate Definition 0 2 1 D 1 1 1 11 E Gleaner 34 3 3 3 3 81 0 7 0 7 0 7 0 49 5y 5 5 25 52 5 5 25 We especially want to call your attention to the last example in each column Note that 5 means that 5 is the base used as a factor twice However 5 means that 5 is the base and after it is squared we take the opposite of the result Properties of Exponents In a previous algebra course you may have seen some properties pertaining to the use of positive integers as exponents Those properties can be summarized as follows PROPERTY 0 1 Properties of Exponents If a and b are real numbers and m and n are positive integers then 1 b pm pn m 22 Ip y e gro 3 ab a b Suc when n gt m b 0 when n m b 0 when n m b 0 Each part o
104. r products contact us at Thomson Learning Academic Resource Center 1 800 423 0563 For permission to use material from this text or product submit a request online at http www thomsonrights com Any additional questions about permissions can be submitted by email to thomsonrights thomson com Library of Congress Control Number 2004113501 ISBN 0 534 99846 1 Instructor s Edition ISBN 0 534 41866 X Art Director Vernon Boes Print Media Buyer Barbara Britton Permissions Editor Chelsea Junget Production Service Susan Graham Cover Designer Roger Knox Cover Image Getty Images Cover Printer Coral Graphic Services Compositor G amp S Typesetters Printer Quebecor World Taunton Thomson Higher Education 10 Davis Drive Belmont CA 94002 3098 USA Asia including India Thomson Learning 5 Shenton Way 301 01 UIC Building Singapore 068808 Australia New Zealand Thomson Learning Australia 102 Dodds Street Southbank Victoria 3006 Australia Canada Thomas Nelson 1120 Birchmount Road Toronto Ontario MIK 5G4 Canada UK Europe Middle East Africa Thomson Learning High Holborn House 50 51 Bedford Row London WC1R 4LR United Kingdom Latin America Thomson Learning Seneca 53 Colonia Polanco 11560 Mexico D F Mexico Spain includes Portugal Thomson Paraninfo Calle Magallanes 25 28015 Madrid Spain Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplic
105. raphing utilities are available we will use mostly generic terminology and let you consult a user s manual for specific key punching instructions We urge you to study the graphing calculator examples in this text even if you do not have access to a graphing utility The examples are chosen to re inforce concepts under discussion Furthermore for those who do have access to a graphing utility we provide Graphing Calculator Activities in many of the prob lem sets Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 12 Chapter O Some Basic Concepts of Algebra A Review Courtesy of Texas Instruments Figure 0 12 Graphing calculators have display windows large enough to show graphs This window feature is also helpful when you re using a graphing calculator for computational purposes because it allows you to see the entries of the problem Figure 0 13 shows a display window for an example of the distributive property Note that we can check to see that the correct numbers and operational symbols have been entered Also note that the answer is given below and to the right of the problem Figure 0 13 Algebraic Expressions Algebraic expressions such as 2x 8xy 3xy 4abc z are called terms A term is an indicated product and may have any number of factors The variables of a term are called literal factors and the numerica
106. rs in standard form 45 3i 7i 46 5i 8i 47 4i 3 2i 48 5i 2 6i 49 3 27 4 6i 50 7 3i 8 4i 51 4 5i 2 9i 52 1 i 2 i 53 2 37 4 6i 54 55 6 47 1 2i 56 3 7i 2 10i 7 37 2 8i 57 3 47 58 4 2i Fom riy 60 2 57 61 8 7i 8 7i 62 5 3i 5 3i 63 2 3i 2 3i 64 6 7i 6 7i H5 Thoughts into words 80 Is every real number also a complex number Ex plain your answer EE Further investigations 82 Observe the following powers of i i V 1 Deed 33 2 rH i 1 i i f P 1 1 1 0 8 Complex Numbers 83 For Problems 65 78 find each quotient and express the answers in standard form 4i 3i E ng 6 o4 2i 2343 3 5i 67 68 37 47 3 fi 69 70 2i 4i 3 27 2 85i Th Fe Si US gam 4 Ti 3 9i 73 74 UE ji gu k gg A 2 di 3 7i 1 4 9i T ae 79 Using a bi and c di to represent two complex numbers verify the following properties a The conjugate of the sum of two complex num bers is equal to the sum of the conjugates of the two numbers b The conjugate of the product of two complex numbers is equal to the product of the conjugates of the numbers 81 Can the product of two nonreal complex numbers be a real number Explain your answer Any power of i greater than 4 can
107. sed to represent numbers The operations of addition subtraction multiplication and division are commonly in dicated by the symbols X and respectively These symbols enable us to form specific numerical expressions For example the indicated sum of 6 and 8 can be written 6 8 In algebra we use variables to generalize arithmetic ideas For example by using x and y to represent any two numbers we can use the expression x y to rep resent the indicated sum of any two numbers The x and y in such an expression are called variables and the phrase x y is called an algebraic expression Many of the notational agreements we make in arithmetic can be extended to algebra with a few modifications The following chart summarizes those nota tional agreements regarding the four basic operations Addition 4 6 de oP iV The sum of x and y Subtraction l TR a Wb The difference of a and b Multiplication 7 X 5or7 5 a b a b a b The product of a and b a b or ab 8 Division 8 4 4 8 4 x y 24 x y The quotient of x divided 4 by y or 4 8 ory x y 0 Note the different ways of indicating a product including the use of parentheses The ab form is the simplest and probably the most widely used form Expressions such as abc 6xy and 14xyz all indicate multiplication Notice the various forms x used to indicate division In algebra the fraction forms 7 and x y are generally used although the other forms
108. sibilities to simplify the resulting fraction 2x 3y Solution Using an LCD of 6xy we can proceed as follows x ay GG G2 2x 3y 2x N3y 3y 2x 9 10x 6xy 6xy _ 9y 10x 6 Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 0 5 Rational Expressions 53 7 11 EXAMPLE 4 btract EZZYYIFXNENM 595 9 FA Solution CE We can factor the numerical coefficients of the denominators into primes to help find the LCD 1200 2 2 3 a b o 2 3 152 3 5 4 j LCD 2 2 3 5 a b 60a b 7 11 7 XS 11 12ab 15a 12ab Sa 15a 4b 35a 440 60a b 600 35a 44b ML 60a b 8 2 add 2 x 4x x Solution E gt 4 P l uidi X LCD x x 4 x x 8 T 2 8 S x x 4 x x x 4 x x 4 8 2 x 4 x x 4 x x 4 _ 8 2x 8 x x 4 28 x x 4 x4 In Figure 0 20 we give some visual support for our answer in Example 5 by 2 2 and Y Certainly their graphs appear to be iden 4x x x 4 graphing Y x 2 7 x Ai Xx tical but a word of caution is needed here Actually the graph of Y Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 54 Chapter O Some Basic Concepts of Algebra A Review
109. ted operations and express the resulting complex numbers in standard form DPD Sl vr ter 23 5 70 45 21 24 7 6i 7 6i 5V6 195 312 21 V 48x y Il ap 27 25 SIT Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User Answers to Odd Numbered Problems and All Chapter Review Chapter Test and Cumulative Review Problems TAM 37 79 39 6 41 19 43 P 1 1 y 8 Problem Set 0 1 page 15 45 47 49 Ski 1 True 3 False 5 False 7 True a a 4 E a i 9 False 11 46 13 0 14 46 y x 2a d 15 V5 V2 17 0 14 19 C ATE m y 97 api T 21 C 23 lt 25 C 27 Z a 29 C 31 33 1 35 0 1 2 3 6L 63 20x y 65 27x y STs I 46 3 2 1 0 1 39 41 0 1 2 86 6 6 43 a 18 c 39 e 35 6r Bw 69 8x 7L 73 35 45 Commutative property of multiplication n 5 1 47 Identity property of multiplication 75 EZ TA EX 79 4 81 7 49 Multiplication property of negative one y 2 ab 5 3 4x y 51 Distributive property 33 gt 85 3b 2a 37 gt 53 Commutative property of multiplication xy ab xy 55 Distributive property 89 12707 91 1 93 lt 95 c4 57 Associative property of multiplication 97 x 99 6 2 10 101 4 12 10 59 22 61 100 63 21 65 8 103 180 000 105 0 0000023 107 0 04 67 19 69
110. tors of the radi cand are perfect nth powers can be simplified as follows ne V28 WANT 2V1 9 x9 3 3 s NV2A N 8N3 23 27 Wor 3 3 Before we consider more examples let s summarize some ideas about sim plifying radicals A radical is said to be in simplest radical form if the following con ditions are satisfied Now let s consider an example in which neither the numerator nor the de nominator of the radicand is a perfect nth power 2 v2 V2 V3 V6 3 3 wg wa 3 Form of 1 The process used to simplify the radical in this example is referred to as rationaliz ing the denominator There is more than one way to rationalize the denominator as illustrated by the next example 8 TES Solution A Vs vs v8 V40 V4V10 2V10 _ V10 vs vs vs 8 8 8 4 Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 66 Chapter O Some Basic Concepts of Algebra A Review Solution B vs VS v2 0 Vio Vio v8 V8 v2 Vie 4 Solution C Vs vs vs _ Vi V2 Vi0 i va waqwe 2V2 2V2 V2 4 The three approaches in Example 1 again illustrate the need to think first and then push the pencil You may find one approach easier than another Simpity 8 V8 Solution V6 6 Va D ars Remember that V8 8 Vb b 3 Reduce the fraction M8 V4 _N3 2 3 simphiy V5 Solution 15 27 3 15
111. ts of Algebra A Review that you understand these properties because they not only facilitate manipula tions with real numbers but also serve as a basis for many algebraic computations The variables a b and c represent real numbers Properties of Real Numbers Closure properties a bis a unique real number ab is a unique real number Commutative properties atb bta ab ba Associative properties se Im ae e ar Wb 3r ab c a bc Identity properties There exists a real number 0 such that a 0 0 a a There exists a real number 1 such that a 1 1 a a Inverse properties For every real number a there exists a unique real number a such that a a a a 0 For every nonzero real number a there exists a unique real number il such that a a a L1 Multiplication property a 0 0 a 0 of zero Multiplication property a 1 1 a a of negative one Distributive property a b c ab ac Let s make a few comments about the properties of real numbers The set of real numbers is said to be closed with respect to addition and multiplication That is the sum of two real numbers is a real number and the product of two real num bers is a real number Closure plays an important role when we are proving addi tional properties that pertain to real numbers Addition and multiplication are said to be commutative operations This means that the order in which yo
112. u add or multiply two real numbers does not af fect the result For example 6 8 8 6 and 4 3 3 4 It is important to realize that subtraction and division are riot commutative operations Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 0 1 Some Basic Ideas 11 order does make a difference For example 3 4 1 but 4 3 1 Likewise 1 2 1 2 but1 2 5 Addition and multiplication are associative operations The associative properties are grouping properties For example 8 9 6 8 9 6 changing the grouping of the numbers does not affect the final sum Likewise for multiplication 4 3 2 4 3 2 Subtraction and division are not associative operations For example 8 6 10 8 but 8 6 10 12 An example showing that division is not associative is 8 4 2 1 but 8 4 2 4 Zero is the identity element for addition This means that the sum of any real number and zero is identically the same real number For example 87 0 0 87 87 One is the identity element for multiplication The product of any real number and 1 is identically the same real number For example 119 1 1 119 119 The real number a is called the additive inverse of a or the opposite of a The sum of a number and its additive
113. umerator Finally let s consider an example involving the root of a root Sinpiity VVZ ED Solution PROBLEM SET 0 7 For Problems 1 16 evaluate 1 4912 2 64 5 3 3235 4 8 5 g5 6 64 1 1 1 2 27 1 3 7 8 1 9 1632 10 0 008 2 3 11 0 01 12 27 13 64756 14 16 4 1 18 1M 5 S E Copyright 2005 Thomson Learning Inc All Rights Reserved V V Qty c aus 2 For Problems 17 32 perform the indicated operations and simplify Express final answers using positive expo nents only 17 3x 5x 18 2x 6x 19 y 5 y 1 20 2x x 19 21 Ax I4 y 125 22 5x Py 24x3 5 18x 2 23 24 X 6x5 9x13 56016 48b Maca AR p puis 801 1283 2734 6x2 5 2 27 S 28 es 3y 7y May not be copied scanned or duplicated in whole or in part Licensed to iChapters User ur gx 29 p y Ante V3 1 2 31 aes 32 5 For Problems 33 48 perform the indicated operations and express the answer in simplest radical form 33 V2W2 34 W3V3 35 Vx Vx 36 Wx Vx 39 Vab N ab 40 Vab N a b 41 W4V8 42 W 9V 27 43 SIS 45 46 o UA a 5 als als Dal Al o NIIN t2 m ON 47 48 Ww E N E EE Thoughts into words 58 Your friend keeps getting an error message when evaluating 4 on his calculator What error is he probably making B5 Further investigations Sometimes we m
114. xpress final answers in 2y V2 5 simplest form 8 24xy ld 21 67 uus 68 e Xy Xy 14a a LULA F 48 3V2 V3 Vx 2Vy 18xly 16y 6p 15ab i y 49 xit 3x 4 33 8x 5 For Problems 69 74 perform the indicated operations x i x Ax and express the answers in simplest radical form a OPE 69 V 5V S 70 Nx 2x 8 6x 13x 5 v EJ sivesi vend 72 V xy V xy f SD 12 di1 a Edd 4 3 4 5 3 VB y 73 74 3 4 2 5 3 vs y 53 5 Fo 54 5 n 5n n x 7x x For Problems 75 86 perform the indicated operations 3x 4 55 I and express the resulting complex number in standard x 6x 40 ux 16 form 56 2 a 75 7 3i 4 9i x 2 x 2 x 4x 76 2 10i 3 8i 77 1 4i 2 6i For Problems 57 59 simplify each complex fraction 78 3i 7i 79 2 5i 3 4i 3 2 TE 80 3 7 6 7i 81 4 27 4 i 57 7 58 82 5 2i 5 2i y ty 5 2 37 83 ai 84 TE 3 m L 4i x h x 1 2i 6i 59 Mu h BIO S a Copyright 2005 Thomson Learning Inc All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 88 Chapter O Some Basic Concepts of Algebra A Review For Problems 87 92 write each in terms of and simplify For Problems 93 and 94 use scientific notation and the roperties of exponents to help with the computations 87 V 100 88 V 40 pn

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