6.4 Dividing Polynomials: Long Division and Synthetic Division

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1 6 CHAPTER 6 Rational Epressions 6. Whih of the following are equivalent to? y a., b. # y. y, y 6. Whih of the following are equivalent to 5? a a. 5, b. a 5, 5. # a a 6. In your own words, eplain one method for simplifying a omple fration. 64. Eplain your favorite method for simplifying a omple fration and why. Simplify y - 5 y - y + 7 y y - y - y + y a a - 7. a + a y y - f a + h - f a In the study of alulus, the differene quotient h is often found and simplified. Find and simplify this quotient for eah funtion f() by following steps a through d. a. Find a + h. b. Find f(a). f a + h - f a. Use steps a and b to find h d. Simplify the result of step. 7. f = 74. f = Dividing Polynomials: Long Division and Syntheti Division S Divide a Polynomial by a Monomial. Divide by a Polynomial. Use Syntheti Division to Divide a Polynomial by a Binomial. 4 Use the Remainder Theorem to Evaluate Polynomials. Dividing a Polynomial by a Monomial Reall that a rational epression is a quotient of polynomials. An equivalent form of a rational epression an be obtained by performing the indiated division. For eample, the rational epression an be thought of as the polynomial divided by the monomial 5. To perform this division of a polynomial by a monomial (whih we do on the net page), reall the following addition fat for frations with a ommon denominator. a + b = a + b If a, b, and are monomials, we might read this equation from right to left and gain insight into dividing a polynomial by a monomial. v Dividing a Polynomial by a Monomial Divide eah term in the polynomial by the monomial. a + b = a + b, where 0

2 Setion 6.4 Dividing Polynomials: Long Division and Syntheti Division 6 EXAMPLE Divide by 5. Solution We divide eah term of by 5 and simplify = = Chek: To hek, see that (quotient) divisor = dividend, or = Divide 8a - a + 0a by 6a. EXAMPLE Divide: 5 y - 5 y - y - 6. y Solution We divide eah term in the numerator by y. 5 y - 5 y - y - 6 y = 5 y y - 5 y y - y y - 6 y = y y Divide: 5a b 4-8a b + ab - 8b ab. Dividing by a Polynomial To divide a polynomial by a polynomial other than a monomial, we use long division. Polynomial long division is similar to long division of real numbers. We review long division of real numbers by dividing 7 into Divisor: = 8. 6 Subtrat and bring down the net digit in the dividend = 4. Subtrat. The remainder is. The quotient is 4 remainder. 7 divisor Chek: To hek, notie that 47 + = 96, the dividend. This same division proess an be applied to polynomials, as shown net. EXAMPLE Divide by +. Solution is the dividend, and + is the divisor. Step. Divide by (Continued on net page) =, so is the first term of the quotient.

3 64 CHAPTER 6 Rational Epressions Step. Multiply Like terms are lined up vertially. Step. Subtrat + 4 from by hanging the signs of + 4 and adding Step 4. Bring down the net term, -0, and start the proess over T -5-0 Step 5. Divide -5 by = -5, so -5 is the seond term of the quotient. Step 6. Multiply Multiply: Like terms are lined up vertially. Step 7. Subtrat by hanging signs of -5-0 and adding Subtrat. Remainder Then = - 5. There is no remainder. + Chek: Chek this result by multiplying - 5 by +. Their produt is = - - 0, the dividend. Divide by +.

4 Setion 6.4 Dividing Polynomials: Long Division and Syntheti Division 65 EXAMPLE 4 Divide: 6-9 +, - 5. Solution T Divide 6 =. Multiply - 5. Subtrat by adding the opposite. Bring down the net term, +. Divide -9 = -. Multiply Subtrat by adding the opposite. Chek: divisor # quotient + remainder v v ( - 5) # ( - ) + (-) = v The division heks, so = or = The dividend Helpful Hint This fration is the remainder over the divisor. 4 Divide by -. EXAMPLE 5 Divide: , + 4. Solution Divide 7 = Subtrat. Bring down. - = -, a term of the quotient Subtrat. Bring down = 50, a term of the quotient Subtrat. Thus, = or Divide , +.

5 66 CHAPTER 6 Rational Epressions EXAMPLE 6 Divide by -. Solution Before dividing, we represent any missing powers by the produt of 0 and the variable raised to the missing power. There is no term in the dividend, so we inlude 0 to represent the missing term. Also, there is no term in the divisor, so we inlude 0 in the divisor T = Subtrat. Bring down -8. =, a term of the quotient Subtrat. Bring down 6. =, a term of the quotient Subtrat. The division proess is finished when the degree of the remainder polynomial is less than the degree of the divisor. Thus, = Divide by +. EXAMPLE 7 Divide by +. Solution We replae the missing terms in the dividend with 0 and Thus, = T Subtrat. Bring down Subtrat. Bring down Divide 64-5 by 4-5. CONCEPT CHECK In a division problem, the divisor is 4-5. The division proess an be stopped when whih of these possible remainder polynomials is reahed? a b Answer to Conept Chek: Using Syntheti Division to Divide a Polynomial by a Binomial When a polynomial is to be divided by a binomial of the form -, a shortut proess alled syntheti division may be used. On the net page, on the left is an eample of long division, and on the right, the same eample showing the oeffiients of the variables only.

6 Setion 6.4 Dividing Polynomials: Long Division and Syntheti Division Notie that as long as we keep oeffiients of powers of in the same olumn, we an perform division of polynomials by performing algebrai operations on the oeffiients only. This shortut proess of dividing with oeffiients only in a speial format is alled syntheti division. To find - - +, - by syntheti division, follow the net eample. EXAMPLE 8 Use syntheti division to divide by -. Solution To use syntheti division, the divisor must be in the form -. Sine we are dividing by -, is. Write down and the oeffiients of the dividend. Net, draw a line and bring down the first oeffiient of the dividend. 6 Multiply # and write down the produt, Add Write down the sum, # 5 = = # = = 7. The quotient is found in the bottom row. The numbers, 5, and are the oeffiients of the quotient polynomial, and the number 7 is the remainder. The degree of the quotient polynomial is one less than the degree of the dividend. In our eample, the degree of the dividend is, so the degree of the quotient polynomial is. As we found when we performed the long division, the quotient is + 5 +, remainder 7 or Use syntheti division to divide by -.

7 68 CHAPTER 6 Rational Epressions When using syntheti division, if there are missing powers of the variable, insert 0s as oeffiients. EXAMPLE 9 Use syntheti division to divide by +. Solution The divisor is +, whih in the form - is - -. Thus, is -. There is no -term in the dividend, so we insert oeffiient of 0. The dividend oeffiients are, -, -, 0, and The dividend is a fourth-degree polynomial, so the quotient polynomial is a thirddegree polynomial. The quotient is with a remainder of. Thus, = Use syntheti division to divide by +. CONCEPT CHECK Whih division problems are andidates for the syntheti division proess? a. + 5, + 4 b. - +, -. y 4 + y -, + d. 5, - 5 Helpful Hint Before dividing by syntheti division, write the dividend in desending order of variable eponents. Any missing powers of the variable should be represented by 0 times the variable raised to the missing power. EXAMPLE 0 If P = , a. Find P() by substitution. b. Use syntheti division to find the remainder when P() is divided by -. Solution a. P = P = = = = 5 Answer to Conept Chek: a and d Thus, P = 5. b. The oeffiients of P() are, -4, 0, and 5. The number 0 is the oeffiient of the missing power of. The divisor is -, so is. h remainder The remainder when P() is divided by - is 5.

8 Setion 6.4 Dividing Polynomials: Long Division and Syntheti Division 69 0 If P = - 5 -, a. Find P by substitution. b. Use syntheti division to find the remainder when P is divided by -. 4 Using the Remainder Theorem to Evaluate Polynomials Notie in the preeding eample that P = 5 and that the remainder when P is divided by - is 5. This is no aident. This illustrates the remainder theorem. Remainder Theorem If a polynomial P is divided by -, then the remainder is P. EXAMPLE Use the remainder theorem and syntheti division to find P4 if P = Solution To find P4by the remainder theorem, we divide Pby - 4. The oeffiients of P are 4, -5, 5, 0, 7, 0, and 0. Also, is 4. h Thus, P4 = 6, the remainder remainder Use the remainder theorem and syntheti division to find P if P = Voabulary, Readiness & Video Chek Martin-Gay Interative Videos See Video 6.4 Wath the setion leture video and answer the following questions. 4. In the leture before Eample, dividing a polynomial by a monomial is ompared to adding two frations. What role does the monomial play in the fration eample?. From Eample, how do you know when to stop your long division?. From Eample, one you ve ompleted the syntheti division, what does the bottom row of numbers mean? What is the degree of the quotient? 4. From Eample 4, given a polynomial funtion P(), under what irumstanes might it be easier/faster to use the remainder theorem to find P rather than substituting the value for and then simplifying?

9 70 CHAPTER 6 Rational Epressions 6.4 Eerise Set Divide. See Eamples and.. 4a + 8a by a by. a5 b + 6a 4 b 4a 4 b 4. 4 y + y - 4y 4y 5. 4 y + 6y - 4y y y y - 4 y 4 y Divide. See Eamples through , + 8. y + 7y + 0, y , , by by , , , , , , - 9. a b, - 0. a b, - Use syntheti division to divide. See Eamples 8 and MIXED Divide. See Eamples y 4 + 8y + 4y 4y 5 y - 5 y + 0y 5 y , , , , y + 9y - y y + 4. y - 8 y y + 75yz + 5 yz, -5 y y 6 - y z + 7 y, -7yz , , For the given polynomial P and the given, use the remainder theorem to find P. See Eamples 0 and. 47. P = ; 48. P = ; 49. P = ; P = ; - 5. P = ; - 5. P = ; - 5. P = ; 54. P = ; 55. P = ; 56. P = ;

10 Setion 6.4 Dividing Polynomials: Long Division and Syntheti Division 7 REVIEW AND PREVIEW Solve eah equation for. See Setions. and = = = = = = 7 9 Fator the following. See Setions 5.5 and y z a y + + y y - y CONCEPT EXTENSIONS Determine whether eah division problem is a andidate for the syntheti division proess. See the Conept Cheks in this setion , , , , a - b 75. In a long division eerise, if the divisor is 9 -, the division proess an be stopped when the degree of the remainder is a. b.. 9 d. 76. In a division eerise, if the divisor is -, the division proess an be stopped when the degree of the remainder is a. b. 0. d. 77. A board of length meters is to be ut into three piees of the same length. Find the length of eah piee. ( 4 6 8) m 80. If the area of a parallelogram is square entimeters and its base is - 7 entimeters, find its height. ( 7) m 8. If the area of a parallelogram is square entimeters and its base is + 5 entimeters, find its height.? Height ( 5) entimeters 8. If the volume of a bo is ubi meters, its height is meters, and its length is + 7 meters, find its width. Divide. Width 8. a b, a b, a b, a b, + meters ( 7) meters , , The perimeter of a regular heagon is given to be miles. Find the length of eah side. 79. If the area of the retangle is square inhes, and its length is 5 + inhes, find its width. For eah given f() and g(), find f. Also find any -values g that are not in the domain of f. (Note: Sine g() is in the g denominator, g() annot be 0.) 89. f = ; g = f = ; g = 9. f = ; g = - 9. f = ; g = +? (5 ) in.

11 7 CHAPTER 6 Rational Epressions 9. Try performing the following division without hanging the order of the terms. Desribe why this makes the proess more ompliated. Then perform the division again after putting the terms in the dividend in desending order of eponents Eplain how to hek polynomial long division. 95. Eplain an advantage of using the remainder theorem instead of diret substitution. 96. Eplain an advantage of using syntheti division instead of long division. We say that is a fator of 8 beause divides 8 evenly, or with a remainder of 0. In the same manner, the polynomial - is a fator of the polynomial beause the remainder is 0 when is divided by -. Use this information for Eerises 97 and Use syntheti division to show that + is a fator of Use syntheti division to show that - is a fator of If a polynomial is divided by - 5, the quotient is and the remainder is. Find the original polynomial. 00. If a polynomial is divided by +, the quotient is and the remainder is -. Find the original polynomial. 0. ebay is the leading online aution house. ebay s annual net profit an be modeled by the polynomial funtion P = , where P is net profit in millions of dollars and is the number of years sine 000. ebay s annual revenue an be modeled by the funtion R = 0-88, where R is revenue in millions of dollars and is years after 000. (Soure: ebay, In., annual reports ) a. Given that net profit Net profit margin = revenue, write a funtion, m, that models ebay s net profit margin. b. Use part (a) to predit ebay s profit margin in 05. Round to the nearest hundredth. 0. Kraft Foods is a provider of many of the best-known food brands in our supermarkets. Among their wellknown brands are Kraft, Osar Mayer, Mawell House, and Oreo. Kraft Foods annual revenues sine 005 an be modeled by the polynomial funtion R = , where R is revenue in billions of dollars and is the number of years sine 005. Kraft Foods net profit an be modeled by the funtion P = , where P is the net profit in billions of dollars and is the number of years sine 005. (Soure: Based on information from Kraft Foods) a. Suppose that a market analyst has found the model P and another analyst at the same firm has found the model R. The analysts have been asked by their manager to work together to find a model for Kraft Foods profit margin. The analysts know that a ompany s profit margin is the ratio of its profit to its revenue. Desribe how these two analysts ould ollaborate to find a funtion m that models Kraft Foods net profit margin based on the work they have done independently. b. Without atually finding m, give a general desription of what you would epet the answer to be. 0. From the remainder theorem, the polynomial - is a fator of a polynomial funtion P if P is what value? 6.5 Solving Equations Containing Rational Epressions Solve Equations Containing Rational Epressions. Helpful Hint The method desribed here is for equations only. It may not be used for performing operations on epressions. Solving Equations Containing Rational Epressions In this setion, we solve equations ontaining rational epressions. Before beginning this setion, make sure that you understand the differene between an equation and an epression. An equation ontains an equal sign and an epression does not. Equation Epression + 6 = equal sign + 6 Solving Equations Containing Rational Epressions To solve equations ontaining rational epressions, first lear the equation of frations by multiplying both sides of the equation by the LCD of all rational epressions. Then solve as usual.