Concepts Examples Use synthetic division to divide ƒ1x2 = 2x3 - 3x + 2 by x - 1, and write the result as ƒ1x2 = g1x2 # q1x2 + r1x2. 1)2 0 -3 2 2 2 -1 2 2 -1 1 (1111)1111* 5 Coefficients of the quotient Remainder 2x3 - 3x + 2 = 1x - 1212x2 + 2x - 12 + 1 (1)1* (11111)11111* 5 ƒ1x = g1x2 # q1x2 + r x2 By the result above, for ƒ1x2 = 2x3 - 3x + 2, ƒ112 = 1. For the polynomial functions ƒ1x2 = x3 + x + 2 and g1x2 = x3 - 1, ƒ1-12 = 0. Therefore, x - 1-12, or x + 1, is a factor of ƒ1x2. Because x - 1 is a factor of g1x2, g112 = 0. The only rational numbers that can possibly be zeros of ƒ1x2 = 2x3 - 9x2 - 4x - 5 are {1, {5, { 1 2 , and { 5 2 . By synthetic division, it can be shown that the only rational zero of ƒ1x2 is 5. 5)2 -9 -4 -5 10 5 5 2 1 1 0 ƒ152 ƒ1x2 = x3 + x + 2 has at least one and at most three distinct zeros. 1 + 2i is a zero of ƒ1x2 = x3 - 5x2 + 11x - 15, and therefore its conjugate 1 - 2i is also a zero. 421 CHAPTER 3 Test Prep 3.2 Synthetic Division Division Algorithm Let ƒ1x2 and g1x2 be polynomials with g1x2 of lesser degree than ƒ1x2 and g1x2 of degree 1 or more. There exist unique polynomials q1x2 and r1x2 such that ƒ1x2 =g1x2 # q1x2 +r1x2, where either r1x2 = 0 or the degree of r1x2 is less than the degree of g1x2. Synthetic Division Synthetic division is a shortcut method for dividing a polynomial by a binomial of the form x - k. Remainder Theorem If a polynomial ƒ1x2 is divided by x - k, the remainder is ƒ1k2. 3.3 Zeros of Polynomial Functions Factor Theorem For any polynomial function ƒ1x2, x - k is a factor of the polynomial if and only if ƒ1k2 = 0. Rational Zeros Theorem If p q is a rational number written in lowest terms, and if p q is a zero of ƒ, a polynomial function with integer coefficients, then p is a factor of the constant term and q is a factor of the leading coefficient. Fundamental Theorem of Algebra Every function defined by a polynomial of degree 1 or more has at least one complex zero. Number of Zeros Theorem A function defined by a polynomial of degree n has at most n distinct zeros. Conjugate Zeros Theorem If ƒ1x2 defines a polynomial function having only real coefficients and if z = a + bi is a zero of ƒ1x2, where a and b are real numbers, then the conjugate z = a - bi is also a zero of ƒ1x2.
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