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Polynomial and Synthetic Division 153 333202_0203.qxd 12/7/05 9:23 AM Page 153 Section 2.3 Polynomial and Synthetic Division 153 2.3 Polynomial and Synthetic Division What you should learn Long Division of Polynomials •Use long division to divide polynomials by other In this section, you will study two procedures for dividing polynomials. These polynomials. procedures are especially valuable in factoring and finding the zeros of polyno- •Use synthetic division to divide mial functions. To begin, suppose you are given the graph of polynomials by binomials of ͑ ͒ ϭ 3 Ϫ 2 ϩ Ϫ the form͑x Ϫ k͒ . f x 6x 19x 16x 4. •Use the Remainder Theorem Notice that a zero of f occurs at x ϭ 2, as shown in Figure 2.27. Because x ϭ 2 and the Factor Theorem. is a zero of f, you know that ͑x Ϫ 2͒ is a factor of f͑x͒. This means that there ͑ ͒ Why you should learn it exists a second-degree polynomial q x such that ͑ ͒ ϭ ͑ Ϫ ͒ ͑ ͒ Synthetic division can help f x x 2 и q x . you evaluate polynomial func- To find q͑x͒, you can use long division, as illustrated in Example 1. tions. For instance, in Exercise 73 on page 160, you will use synthetic division to determine Example 1 Long Division of Polynomials the number of U.S. military personnel in 2008. Divide 6x3 Ϫ 19x 2 ϩ 16x Ϫ 4 by x Ϫ 2, and use the result to factor the polyno- mial completely. Solution 6x3 Think ϭ 6x2. x Ϫ7x2 Think ϭϪ7x. x 2x Think ϭ 2. x 6 x2 Ϫ 7 x ϩ 2 x Ϫ 2 ) 6x3 Ϫ 19 x2 ϩ 16 x Ϫ 4 ©Kevin Fleming/Corbis 6 x3 Ϫ 12 x2 Multiply: 6x2͑x Ϫ 2͒. Ϫ7x2 ϩ 16 x Subtract. Ϫ7x2 ϩ 14 x Multiply: Ϫ7x͑x Ϫ 2͒. 2 x Ϫ 4 Subtract. y 2 x Ϫ 4 Multiply: 2͑x Ϫ 2͒. 1 , 0 1 ()2 0 Subtract. 2 , 0 ()3 x From this division, you can conclude that 13 6x3 Ϫ 19x 2 ϩ 16x Ϫ 4 ϭ ͑x Ϫ 2͒͑6x2 Ϫ 7x ϩ 2͒ −1 and by factoring the quadratic 6x2 Ϫ 7x ϩ 2, you have −2 6x3 Ϫ 19x 2 ϩ 16x Ϫ 4 ϭ ͑x Ϫ 2͒͑2x Ϫ 1͒͑3x Ϫ 2͒. − 3 − 2 − Note that this factorization agrees with the graph shown in Figure 2.27 in that the 3 f(x) = 6x 19x + 16x 4 ϭ ϭ 1 ϭ 2 three x-intercepts occur at x 2, x 2, and x 3. FIGURE 2.27 Now try Exercise 5. 333202_0203.qxd 12/7/05 9:23 AM Page 154 154 Chapter 2 Polynomial and Rational Functions Note that one of the many uses of In Example 1,x Ϫ 2 is a factor of the polynomial 6x3 Ϫ 19x 2 ϩ 16x Ϫ 4, polynomial division is to write a and the long division process produces a remainder of zero. Often, long division function as a sum of terms to find slant 2 ϩ ϩ asymptotes (see Section 2.6).This is a will produce a nonzero remainder. For instance, if you divide x 3x 5 by skill that is also used frequently in x ϩ 1, you obtain the following. calculus. x ϩ 2 Quotient Divisorx ϩ 1 ) x2 ϩ 3 x ϩ 5 Dividend x2 ϩ x 2 x ϩ 5 2 x ϩ 2 3 Remainder In fractional form, you can write this result as follows. Remainder Dividend Quotient x 2 ϩ 3x ϩ 5 3 ϭ x ϩ 2 ϩ x ϩ 1 x ϩ 1 Divisor Divisor Have students identify the dividend, This implies that divisor, quotient, and remainder when 2 ͑ ϩ ͒ dividing polynomials. x ϩ 3x ϩ 5 ϭ ͑x ϩ 1͒(x ϩ 2͒ ϩ 3 Multiply each side by x 1 . which illustrates the following theorem, called the Division Algorithm. The Division Algorithm If f ͑x͒ and d͑x͒ are polynomials such that d͑x͒ 0, and the degree of d͑x͒ is less than or equal to the degree of f ͑x͒, there exist unique polynomials q͑x͒ and r͑x͒ such that f ͑x͒ ϭ d͑x͒q͑x͒ ϩ r͑x͒ Dividend Quotient Divisor Remainder where r͑x͒ ϭ 0 or the degree of r͑x͒ is less than the degree of d͑x͒. If the remainder r͑x͒ is zero, d͑x͒ divides evenly into f ͑x͒. The Division Algorithm can also be written as f ͑x͒ r͑x͒ ϭ q͑x͒ ϩ . d͑x͒ d͑x͒ In the Division Algorithm, the rational expression f ͑x͒͞d͑x͒ is improper because the degree of f ͑x͒ is greater than or equal to the degree of d͑x͒. On the other hand, the rational expression r͑x͒͞d͑x͒ is proper because the degree of r͑x͒ is less than the degree of d͑x͒. 333202_0203.qxd 12/7/05 9:23 AM Page 155 Section 2.3 Polynomial and Synthetic Division 155 Before you apply the Division Algorithm, follow these steps. 1. Write the dividend and divisor in descending powers of the variable. 2. Insert placeholders with zero coefficients for missing powers of the variable. Example 2 Long Division of Polynomials Divide x3 Ϫ 1 by x Ϫ 1. Solution Because there is no x2-term or x-term in the dividend, you need to line up the subtraction by using zero coefficients (or leaving spaces) for the missing terms. x2 ϩ x ϩ 1 x Ϫ 1 ) x 3 ϩ 0 x2 ϩ 0 x Ϫ 1 x 3 Ϫ x 2 x2 ϩ 0 x x2 Ϫ x x Ϫ 1 x Ϫ 1 0 So,x Ϫ 1 divides evenly into x3 Ϫ 1, and you can write x3 Ϫ 1 ϭ x2 ϩ x ϩ 1, x 1. x Ϫ 1 Now try Exercise 13. You can check the result of Example 2 by multiplying. ͑x Ϫ 1͒͑x2 ϩ x ϩ 1͒ ϭ x3 ϩ x2 ϩ x Ϫ x2 Ϫ x Ϫ 1 ϭ x3 Ϫ 1 Example 3 Long Division of Polynomials Divide 2x4 ϩ 4x3 Ϫ 5x 2 ϩ 3x Ϫ 2 by x 2 ϩ 2x Ϫ 3. Solution 2 x2 ϩ 1 x2 ϩ 2x Ϫ 3 ) 2x 4 ϩ 4 x 3 Ϫ 5 x2 ϩ 3 x Ϫ 2 2x 4 ϩ 4 x 3 Ϫ 6 x2 x2 ϩ 3 x Ϫ 2 x2 ϩ 2 x Ϫ 3 x ϩ 1 Note that the first subtraction eliminated two terms from the dividend. When this happens, the quotient skips a term. You can write the result as Remind students that when division 2x4 ϩ 4x3 Ϫ 5x 2 ϩ 3x Ϫ 2 x ϩ 1 yields a remainder, it is important that ϭ 2x 2 ϩ 1 ϩ . they write the remainder term correctly. x 2 ϩ 2x Ϫ 3 x 2 ϩ 2x Ϫ 3 Now try Exercise 15. 333202_0203.qxd 12/7/05 9:23 AM Page 156 156 Chapter 2 Polynomial and Rational Functions Synthetic Division There is a nice shortcut for long division of polynomials when dividing by divisors of the form x Ϫ k. This shortcut is called synthetic division. The pattern for synthetic division of a cubic polynomial is summarized as follows. (The pattern for higher-degree polynomials is similar.) Synthetic Division (for a Cubic Polynomial) To divide ax3 ϩ bx2 ϩ cx ϩ d by x Ϫ k, use the following pattern. Point out to students that a graphing k abcd Coefficients of dividend utility can be used to check the answer ka to a polynomial division problem.When Vertical pattern: Add terms. students graph both the original Diagonal pattern: Multiply by k. polynomial division problem and the a r Remainder answer in the same viewing window, the graphs should coincide. Coefficients of quotient Synthetic division works only for divisors of the form x Ϫ k. [Remember that x ϩ k ϭ x Ϫ ͑Ϫk͒.]You cannot use synthetic division to divide a polynomial by a quadratic such as x 2 Ϫ 3. Example 4 Using Synthetic Division Use synthetic division to divide x 4 Ϫ 10x 2 Ϫ 2x ϩ 4 by x ϩ 3. Solution You should set up the array as follows. Note that a zero is included for the missing x3-term in the dividend. Ϫ3 10Ϫ10 Ϫ2 4 Then, use the synthetic division pattern by adding terms in columns and multi- plying the results by Ϫ3. Divisor:x ϩ 3 Dividend: x 4 Ϫ 10x2 Ϫ 2x ϩ 4 Ϫ31 0 Ϫ10 Ϫ2 4 Ϫ3 9 3 Ϫ3 1 Ϫ3 Ϫ1 1 1 Remainder: 1 Quotient: x3 Ϫ 3x2 Ϫ x ϩ 1 So, you have x4 Ϫ 10x2 Ϫ 2x ϩ 4 1 ϭ x3 Ϫ 3x2 Ϫ x ϩ 1 ϩ . x ϩ 3 x ϩ 3 Now try Exercise 19. 333202_0203.qxd 12/7/05 9:23 AM Page 157 Section 2.3 Polynomial and Synthetic Division 157 The Remainder and Factor Theorems The remainder obtained in the synthetic division process has an important interpretation, as described in the Remainder Theorem. The Remainder Theorem If a polynomial f ͑x͒ is divided by x Ϫ k, the remainder is r ϭ f ͑k͒. For a proof of the Remainder Theorem, see Proofs in Mathematics on page 213. The Remainder Theorem tells you that synthetic division can be used to evaluate a polynomial function. That is, to evaluate a polynomial function f ͑x͒ when x ϭ k, divide f ͑x͒ by x Ϫ k. The remainder will be f ͑k͒, as illustrated in Example 5.
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