Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition)

Published by Pearson
ISBN 10: 0-32193-104-1
ISBN 13: 978-0-32193-104-7

Chapter 3 - Polynomial and Rational Functions - Section 3.3 Complex Zeros; Fundamental Theorem of Algebra - 3.3 Assess Your Understanding - Page 232: 45



Work Step by Step

The Conjugate Pairs Theorem states that when a polynomial has real coefficients, then any complex zeros occur in conjugate pairs. This means that, when $(p +i \ q)$ is a zero of a polynomial function with a real number of the coefficients, then its conjugate $(p –i q)$, is also a zero of the function. We see that $2+i$ is zero of the polynomial function with real coefficients. This means that $2-i$ is also a zero of the function by the Conjugate Pairs Theorem. This implies that the polynomial function has at least $4$ zeros. However, the function does not have a degree of $4$, it has a degree of $3$. Thus we have a contradiction. Thus, the given statement is false.
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