342 CHAPTER 3 Polynomial and Rational Functions Synthetic division provides an efficient process for dividing a polynomial in x by a binomial of the form x - k. Begin by writing the coefficients of the polynomial in decreasing powers of the variable, using 0 as the coefficient of any missing powers. The number k is written to the left in the same row. The answer is found in the bottom row with the remainder farthest to the right and the coefficients of the quotient on the left when written in order of decreasing degree. CAUTION To avoid errors, use 0 as the coefficient for any missing terms, including a missing constant, when setting up the division. The result of the division in Example 1 can be written as 5x3 - 6x2 - 28x - 2 = 1x + 2215x2 - 16x + 42 + 1-102. Multiply by x + 2. The theorem that follows is a generalization of this product form. EXAMPLE 1 Using Synthetic Division Use synthetic division to perform the division. 5x3 - 6x2 - 28x - 2 x + 2 SOLUTION Express x + 2 in the form x - k by writing it as x - 1-22. -2)5 -6 -28 -2 Coefficients of the polynomial Bring down the 5, and multiply: -2152 = -10. -2)5 -6 -28 -2 -10 5 Add -6 and -10 to obtain -16. Multiply: -21-162 = 32. -2)5 -6 -28 -2 -10 32 5 -16 Add -28 and 32, obtaining 4. Multiply: -2142 = -8. -2)5 -6 -28 -2 -10 32 -8 5 -16 4 Add -2 and -8 to obtain -10. -2)5 -6 -28 -2 -10 32 -8 5 -16 4 -10 Remainder (11+111)1111+1* Quotient Because the divisor x - k has degree 1, the degree of the quotient will always be one less than the degree of the polynomial to be divided. 5x3 - 6x2 - 28x - 2 x + 2 = 5x2 - 16x + 4 + -10 x + 2 S Now Try Exercise 15. Add columns. Be careful with signs. Remember to add remainder divisor . x + 2 leads to -2.
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