Precalculus (10th Edition)

Published by Pearson
ISBN 10: 0-32197-907-9
ISBN 13: 978-0-32197-907-0

Chapter 4 - Polynomial and Rational Functions - 4.6 Complex Zeros; Fundamental Theorem of Algebra - 4.6 Assess Your Understanding - Page 240: 20



Work Step by Step

The Conjugate Pairs Theorem says that if a polynomial has real coefficients, then any complex zeros occur in conjugate pairs. That is, if $a + bi$ is a zero then so is $a – bi$ and vice-versa. Therefore the zeros of the function are: $2\pm i$, $4\pm i$ and $\pm i$ Hence, the function could be (by setting $a=1$): $[x-(2+i)][x-(2-i)](x-i)[x-(-i)][x-(4-i)][x-(4+i)]\\ =(x-2-i)(x-2+i)(x-i)(x+i)(x-4+i)(x-4-i)\\ =x^6-12x^5+55x^4-120x^3+139x^2-108x+85$
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