Section A-2 Polynomials: Basic Operations
Contents
1. 3b m 3n m 8n m 6n m 4n y 2 y I y 3 y 4 2m n 42 3a 2b Problems 43 50 are calculus related Perform the indicated erations and simplify 5Sa h 4 5x 4 44 6 x h 2 6x 2 3 x h 2x h 3x7 2x A x hy 5 x A 4x7 5x 2 x hY 30 h 7 2x7 3x 7 x h 4 x h 9 x 4x 9 x hp x Q x hy 3 h 2x 3x Subtract the sum of the first two polynomials from the sum of the last two 3m 2m 5 4m m 2 3m 3m 2 m m 2 Subtract the sum of the last two polynomials from the sum of the first two 2x7 4xy y 3xy y x7 2xy y x 3xy 2y A 22 Appendix A A BASIC ALGEBRA REVIEW ee In Problems 53 56 perform the indicated operations and simplify 53 2x 2 2 307 2 4 54 2x 1P 2 2x 1 3 2x 1 7 55 3x x x x 2 x 2 Q 3 56 2 x 3 2x 1 x 3 x x 2 57 Show by example that in general a b b Discuss possible conditions on a and b that would make this a valid equation 58 Show by example that in general a b a b Discuss possible conditions on a and b that would make this a valid equation 59 If you are given two polynomials one of degree m and the other of degree n2. D a b c a b c Section A 2 Polynomials 54 55 56 Sa A 2 Polynomials Basic Operations 4 A 11 fe 53 IfA 1 2 3 4 and B 2 4 6 find A x x GA orx EB B x x A and x B If F 2 0 2 and G 1 0 1 2 find A x x Forx G B x x E Fandx G Ifc 0 151515 then 100c 15 1515 and 100c c 15 1515 0 151515 99c 15 15 _ 5 C 99 33 Proceeding similarly convert the repeating decimal 0 090909 into a fraction All repeating decimals are rational numbers and all rational numbers have repeating decimal representations Repeat Problem 55 for 0 181 818 To see how the distributive property is behind the mechan ics of long multiplication compute each of the following and compare Long Use of the Multiplication Distributive Property 23 23712 X12 23 2 10 23 2 23 10 For a and b real numbers justify each step using a prop erty in this section Statement Reason 1 a b a a a b 1 2 a a b 2 3 0 b 3 4 b 4 Bas c Operations Natural Number Exponents Polynomials Combining Like Terms Addition and Subtraction Multiplication Combined Operations Application A 12 Appendix A A BASIC ALGEBRA REVIEW In this section we review the basic operations on polynomials a mathematical form encountered frequently throughout mathematics We start the discu
3. 2 6 x 2 To speed up the process we combine the inner and outer product mentally A 18 L Appendix A A BASIC ALGEBRA REVIEW Products of certain binomial factors occur so frequently that it is useful to remember formulas for their products The following formulas are easily verified by multiplying the factors on the left using the FOIL method SPECIAL PRODUCTS 1 a b a b g b 2 a bF a 2ab bd 3 a bP a 2ab b Explore Discuss A Explain the relationship between special product formula 1 and the areas of the rectangles in the figures a bya b bs B Construct similar figures to provide geometric interpretations for special product formulas 2 and 3 EXAMPLE Multiplying Binomials 6 Multiply D m 2n m 4mn 4n MATCHED PROBLEM Multiply 6 A 4u 3v 2u v B xy 3 2xy 3 C m 4n m 4n D 2u 3v 6x y A 2 Polynomials Basic Operations 4 A 19 Remember to include the sum of the inner and outer terms when using CAUTION the FOIL method to square a binomial That is Qxt 3P x 9 x 3 x 4 6x49 Combined Operations We now consider several examples that use all the operations just discussed Before considering these examples it is useful to summarize order of operation conventions pertaining to exponents multiplication and division and addition and subtraction ORDER OF OPERATIONS 1 Simplify inside th
4. B All rational numbers are real numbers C All natural numbers are rational numbers 45 Give an example of a rational number that is not an integer 46 Give an example of a real number that is not a rational number 47 Given the sets of numbers N natural numbers Z inte gers Q rational numbers and R real numbers indicate to which set s each of the following numbers belongs A 3 B 3 14 C r D 2 48 Given the sets of numbers N Z Q and R see Problem 47 indicate to which set s each of the following num bers belongs AV8 B V2 1414 D gt In Problems 49 and 50 use a calculator to express each number as a decimal fraction to the capacity of your calculator refer to the user s manual for your calculator Observe the repeating decimal representation of the rational numbers and the apparent nonrepeating decimal representation of the irrational numbers BA vs B VI Ox 51 Indicate true T or false F and for each false statement find real number replacements for a and b that will pro vide a counterexample For all real numbers a and b Aa b b a B a b b a C ab ba Dj a b b a 49 A 5 D F 50 A D 58 52 Indicate true T or false F and for each false statement find real number replacements for a b and c that will pro vide a counterexample For all real numbers a b and c A a b c at b t c B a b c a b Cc C a bc ab c
5. m gt n what is the degree of the sum 60 What is the degree of the product of the two polynomials in Problem 59 61 How does the answer to Problem 59 change if the two polynomials can have the same degree 62 How does the answer to Problem 60 change if the two polynomials can have the same degree APPLICATIONS 63 Geometry The width of a rectangle is 5 centimeters less than its length If x represents the length write an alge braic expression in terms of x that represents the perimeter of the rectangle Simplify the expression 64 Geometry The length of a rectangle is 8 meters more than its width If x represents the width of the rectangle write an algebraic expression in terms of x that represents its area Change the expression to a form without parentheses x 65 x 66 67 68 Coin Problem A parking meter contains nickels dimes and quarters There are 5 fewer dimes than nickels and 2 more quarters than dimes If x represents the number of nickels write an algebraic expression in terms of x that represents the value of all the coins in the meter in cents Simplify the expression Coin Problem A vending machine contains dimes and quarters only There are 4 more dimes than quarters If x represents the number of quarters write an algebraic ex pression in terms of x that represents the value of all the coins in the vending machine in cents Simplify the expression Packaging A spherical plasti
6. ERTIES 1 a b c b c a ab ac Vo a ae Gar 88 Se IP Sle ae E ae O88 Sr oh Two terms in a polynomial are called like terms if they have exactly the same variable factors to the same powers The numerical coefficients may or may not be the same Since constant terms involve no variables all constant terms are like terms If a polynomial contains two or more like terms these terms can be com bined into a single term by making use of distributive properties Consider the following example 5x y 2xy xy 2x y 5x y xy Ixy 2xy Sx y x y 2x y 2xy 5 1 2 x y 2xy It should be clear that free use has been made of the real number properties discussed earlier The steps done in the dashed box are usually done mentally and the process is quickly mechanized as follows A 2 Polynomials Basic Operations 4 A 15 Like terms in a polynomial are combined by adding their numerical coefficients EXAMPLE Simplifying Polynomials 2 Remove parentheses and combine like terms A 2Gx 25 5 3x 7 23x 2x 5 1Q 3x 7 Think 6X 4x 10 x 3x 7 7x x 3 B x 2x 6 2x x 2x 3 108 2x 6 DQ x x 2x 3 Be careful with Think the sign here x 2x 6 24 x 2x 3 x 4x 3 C Bx 2x DI 1 Be 2x 1 Remove inner parentheses f
7. c container for designer wristwatches has an inner radius of x centimeters see the figure If the plastic shell is 0 3 centimeters thick write an algebraic expression in terms of x that represents the volume of the plastic used to construct the container Sim plify the expression Recall The volume V of a sphere of radius r is given by V 4ar Packaging A cubical container for shipping computer components is formed by coating a metal mold with poly styrene If the metal mold is a cube with sides x centime ters long and the polystyrene coating is 2 centimeters thick write an algebraic expression in terms of x that rep resents the volume of the polystyrene used to construct the container Simplify the expression Recall The volume V of a cube with sides of length ris given by V Section A 3 Polynomials Factoring Factoring What Does It Mean V Common Factors and Factoring by Grouping Factoring Second Degree Polynomials More Factoring Factoring What Does It Mean A factor of a number is one of two or more numbers whose product is the given number Similarly a factor of an algebraic expression is one of two or more alge braic expressions whose product is the given algebraic expression For example
8. e innermost grouping first then the next innermost and so on 2 3 4 2 338 x 4 33 Wb a2 2 Unless grouping symbols indicate otherwise apply exponents before multiplication or division is performed Ax 2 207 4x 4 2x 8x 8 3 Unless grouping symbols indicate otherwise perform multiplication and division before addition and subtraction In either case proceed from left to right S eS bee On ella EXAMPLE Combined Operations 7 Perform the indicated operations and simplify A 3x 5 3 x x3 x 3x 5 3 x 3x xl 3x 5 3 2x x 3x 5 6x 3x 3x 5 6x 3x 34 34 5 B x 2y 2x 3y 2x y 2x 3xy 4xy 6y 4x7 Axy y 5 2x xy 6y 4x 4xy y 2x 5xy Ty A 20 L Appendix A A BASIC ALGEBRA REVIEW 2m 3n 2m 3n 2m 3n 2m 3n 4m 12mn 9n 8m 24m n 18mn 12mn 36mn 27n 8m 36m n 54mn 270 MATCHED PROBLEM Perform the indicated operations and simplify 7 A 2t 7 2 t 4 l B u 3v 2u v 2u v 4x yy Application EXAMPLE Volume of a Cylindrical Shell Ely A plastic water pipe with a hollow center is 100 inches long 1 inch thick and has an inner radius of x inches see the figure Write an algebraic expression in terms o
9. f the following are polynomials x 3x 2x 1 y x 2xy y 5 xe 2 B Given the polynomial 3x 6x 5 what is the degree of the first term The second term The whole polynomial C Given the polynomial 6x y 3xy what is the degree of the first term The second term The whole polynomial A 14 L Appendix A A BASIC ALGEBRA REVIEW In addition to classifying polynomials by degree we also call a single term polynomial a monomial a two term polynomial a binomial and a three term polynomial a trinomial Sy Monomial x 47 Binomial xt V2 9 Trinomial Combining Like Terms We start with a word about coefficients A constant in a term of a polynomial including the sign that precedes it is called the numerical coefficient or simply the coefficient of the term If a constant doesn t appear or only a sign appears the coefficient is understood to be 1 If only a sign appears the coefficient is understood to be 1 Thus given the polynomial 24 40 97 x 5 2x 4 x 1x 1 xK 5 the coefficient of the first term is 2 the coefficient of the second term is 4 the coefficient of the third term is 1 the coefficient of the fourth term is 1 and the coefficient of the last term is 5 At this point it is useful to state two additional distributive properties of real numbers that follow from the distributive properties stated in Section A 1 ADDITIONAL DISTRIBUTIVE PROP
10. f x that represents the volume of the plastic used to construct the pipe Simplify the expression Recall The volume V of a right circular cylin der of radius r and height h is given by V ah 1 inch x inch 100 inches Solution A right circular cylinder with a hollow center is called a cylindrical shell The volume of the shell is equal to the volume of the cylinder minus the volume of the hole Since the radius of the hole is x inches and the pipe is inch thick the radius of the cylinder is x 1 inches Thus we have Volume of _ Volume of _ Volume of shell cylinder hole Volume m x 1 100 ax 100 100m x 2x 1 100mx 100mx 2007x 100m 100rx 200rx 1007 A 2 Polynomials Basic Operations A 21 MATCHED PROBLEM 8 A plastic water pipe is 200 inches long 2 inches thick and has an outer radius of x inches Write an algebraic expression in terms of x that represents the vol ume of the plastic used to construct the pipe Simplify the expression Answers to Matched Problems 2 1 A 32 2x 1 2 2xy y B 5 3 5 C 6 4 6 2 A 4u v B m 3m 2m 4 O x 3x 2 3 3x xX 5x7 2x 2 4 3x 5x 10 5 4 13x 6 6 A 81 2uv 3 B 4y 9 C m 16n D 4u 12uv 9y 36x 12xy y 7 A 2 4 7 B 3 6uv 10v C 647 48x7y 12 y 8 Volume 200m
11. irst 37 2x 1 xYr4 1 2x 2x MATCHED PROBLEM Remove parentheses and combine like terms 2 A 30 2v u Sv B m 3m m 1 2m m 3 O x 2 2x7 Bx 4 Addition and Subtraction Addition and subtraction of polynomials can be thought of in terms of removing parentheses and combining like terms as illustrated in Example 2 Horizontal and vertical arrangements are illustrated in the next two examples You should be able to work either way letting the situation dictate the choice EXAMPLE Adding Polynomials 3 Add x 3x8 2 x 2x 3x and 3x7 4x 5 A 16 L Appendix A A BASIC ALGEBRA REVIEW Solution Add horizontally x 3x x7 x 2x7 3x Gx 4x 5 x 3h 4 x 2x 3x 3x 4x 5 xt 4427 x 5 Or vertically by lining up like terms and adding their coefficients Be a De a ax 3x 44 5 xt 4 2x7 x 5 MATCHED PROBLEM Add horizontally and vertically 3 3xt 20 4x7 x 2x 5x and xr 7x 2 EXAMPLE Subtracting Polynomials 4 Subtract 4x 3x 5 from xr 8 Solution Q 8 4x 3x 5 or x 8 8 47 4 3x 5 4x 3x 5 lt Change signs and add 3x 3x 13 3x 3x 13 MATCHED PROBLEM Subtract 4 2x 5x 4 from 5x 6 When you use a hor
12. izontal arrangement to subtract a polynomial with more than one term you must enclose the polynomial in parentheses Thus to subtract 2x 5 from 4x 11 you must write CAUTION 4x 11 2x 5 and not 4x 11 2x 5 Multiplication Multiplication of algebraic expressions involves the extensive use of distributive properties for real numbers as well as other real number properties A 2 Polynomials Basic Operations 4 A 17 EXAMPLE Multiplying Polynomials 5 Multiply 2x 3 3x 2x 3 Solution 2x 3 3x 2x 3 2x 3x 2x 3 3 3x 2x 3 or using a vertical arrangement ax 2x 3 2x 3 Gx 43 6x 9x7 6x 9 6x 13x 12x 9 MATCHED PROBLEM Multiply 5 2x 3 2x7 3x 2 Thus to multiply two polynomials multiply each term of one by each term of the other and combine like terms Products of certain binomial factors occur so frequently that it is useful to develop procedures that will enable us to write down their products by inspec tion To find the product 2x 1 3x 2 we will use the popular FOIL method We multiply each term of one factor by each term of the other factor as follows F O First Outer Inner Last product product product product J 2x 1 3x 2 6x 4x 3x 2 The inner and outer products are like terms and hence combine into one term Thus 2x 1 3x
13. ng constants and terms of the form ax where a is a real num ber and n is a natural number A polynomial in two variables x and y is con structed by adding and subtracting constants and terms of the form ax y where a is areal number and m and n are natural numbers Polynomials in three or more variables are defined in a similar manner Polynomial forms can be classified according to their degree If a term in a polynomial has only one variable as a factor then the degree of that term is the power of the variable If two or more variables are present in a term as factors then the degree of the term is the sum of the powers of the variables The degree of a polynomial is the degree of the nonzero term with the highest degree in the polynomial Any nonzero constant is defined to be a polynomial of degree 0 The number 0 is also a polynomial but is not assigned a degree Polynomials and Nonpolynomials A Polynomials in one variable vr 3x4 2 6x2 V2x B Polynomials in several variables 3x Ixy y 4x y V3xy z C Nonpolynomials 3 a ae V2x 5 a Vx 3x 1 x KS D The degree of the first term in 6x V2x 4 is 3 the degree of the second term is 1 the degree of the third term is 0 and the degree of the whole polynomial is 3 The degree of the first term in 4x y V3xy is 5 the degree of the sec ond term is 3 and the degree of the whole polynomial is 5 A Which o
14. ssion with a brief review of natural number exponents Integer and rational exponents and their properties will be discussed in detail in subsequent sections Natural Number Exponents The definition of a natural number exponent is given below NATURAL NUMBER EXPONENT DEFINITION For n a natural number and a any real number 1 a arate a Po2 2 2 2 n factors of a 4 factors of 2 Also the first property of exponents is stated as follows FIRST PROPERTY OF EXPONENTS For any natural numbers m and n and any real number a THEOREM 1 a a gie 3x 2x 3 Z 2 x 7 6x Polynomials Algebraic expressions are formed by using constants and variables and the alge braic operations of addition subtraction multiplication division raising to pow ers and taking roots Some examples are We 5 5x1 2x 7 x y T7 2x y c 5 i 1 xX 2x 5 1 1 x An algebraic expression involving only the operations of addition subtraction multiplication and raising to natural number powers on variables and constants is called a polynomial Some examples are 2k 23 A 36 7 x 2y 66 2 Te 5 x 3xy 4y 0 L 3x y xy 2y EXAMPLE 1 MATCHED PROBLEM 1 A 2 Polynomials Basic Operations A 13 In a polynomial a variable cannot appear in a denominator as an exponent or within a radical Accordingly a polynomial in one variable x is constructed by adding or subtracti
15. x 200a x 2 800ax 8007 EXERCISE A 2 30 m m m m 1 31 2 3 a 4 1 a 5 a AL 32 5b 3 2 4 2b 1 2 2 3b Problems 1 8 refer to the following polynomials aay x Fe Qe 34S a 2x 3x7 x4 5 b 2x7 x 1 3x 2 1 What is the degree of a 2 What is the degree of b 35 3 Add a and b 4 Add b and c 36 5 Subtract b from a 6 Subtract c from b 37 7 Multiply a and c 8 Multiply b and c 38 In Problems 9 28 perform the indicated operations and 39 simplify 40 9 2 x 1 3 2x 3 4x 5 Al 10 2 u 1 Gu 2 2 2u 3 11 2y 3y 4 X y 1 12 4a 2a 5 3 a 2 cf 13 m n m n 14 a bya b op 15 4t 3 t 2 16 3x 5 2x 1 43 17 3x 2y x 3y 18 2x 3y x 2y 45 19 2m 7 2m 7 20 3y 2 3y 2 46 21 6x 4y 5x 3y 22 3m 7n 2m 5n 47 23 3x 2y 3x 2y 24 4m 3n 4m 3n 48 25 4x y 26 Bu 4v 49 27 a ba ab b 28 a bya ab b 50 51 B IMMM In Problems 29 42 perform the indicated operations and simplify 29 2x 3 x 2 x 5 1 34 52 2xy yE 2xy y K hk PX hk R n 2n 1 n 4n 3 2x 1 3x 2 3x 2 3a b 3a b 2a
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