352 CHAPTER 3 Polynomial and Rational Functions Conjugate Zeros Theorem The following properties of complex conjugates are needed to prove the conjugate zeros theorem. We use a simplified notation for conjugates here. If z = a + bi, then the conjugate of z is written z, where z = a - bi. For example, if z = -5 + 2i, then z = -5 - 2i. Properties of Conjugates For any complex numbers c and d, the following properties hold true. c +d =c +d, c # d =c # d, and cn = 1c2n In general, if z is a zero of a polynomial function with real coefficients, then so is z. For example, the remainder theorem can be used to show that both 2 + i and 2 - i are zeros of ƒ1x2 = x3 - x2 - 7x + 15. Conjugate ZerosTheorem 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. Proof Start with the polynomial function ƒ1x2 = an x n + a n-1 x n-1 + g+ a 1 x + a0, where all coefficients are real numbers. If the complex number z is a zero of ƒ1x2, then we have the following. ƒ1z2 = anz n + a n-1z n-1 + g+ a 1z + a0 = 0 Take the conjugate of both sides of this equation. anz n + a n-1z n-1 + g+ a 1z + a0 = 0 anz n + a n-1z n-1 + g+ a 1z + a0 = 0 Use generalizations of the properties c + d = c + d and c # d = c # d. an z n + a n-1 z n-1 + g+ a 1 z + a0 = 0 an1z2n + a n-11z2n-1 + g+ a 11z2 + a0 = 0 Use the property cn = 1c2n and the fact that for any real number a, ƒ 1z2 = 0 a = a. Hence z is also a zero of ƒ1x2, which completes the proof. CAUTION When the conjugate zeros theorem is applied, it is essential that the polynomial have only real coefficients. For example, ƒ1x2 = x - 11 + i2 has 1 + i as a zero, but the conjugate 1 - i is not a zero. NOTE In Example 4(a), we cannot clear the denominators in ƒ1x2 by multiplying each side by 2 because the result would equal 2 # ƒ1x2, not ƒ1x2.
RkJQdWJsaXNoZXIy NjM5ODQ=