College Algebra (10th Edition)

Published by Pearson
ISBN 10: 0321979478
ISBN 13: 978-0-32197-947-6

Chapter 5 - Section 5.6 - Complex Numbers; Quadratic Equations in the Complex Number System - 5.6 Assess Your Understanding: 47

Answer

The fourth zero is a real number because complex numbers come in conjugate pairs. Having another complex zero means it will come with a pair, giving the function five zeros, which is not possible.

Work Step by Step

RECALL: The Conjugate Pairs Theorem states that if $a+bi$ is a zero of a polynomial function with real coefficients, then $a-bi$ is also a zero of the function. The function's degree is four so it has four zeros. Having $2, 1+2i,$ and $1-2i$ as zeros mean that the fourth zero must be a real number because complex zeros come in pairs.
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