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 231: 18

Answer

$x^4-2x^3+6x^2-2x+5$

Work Step by Step

Let us consider that $a$ is a zero of a function with multiplicity $b$. Then this factor of the function can be expressed as: $(x-a)^b$. 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 are given that the zeros of the function are: $1\pm2i$ and $\pm i$. Therefore, we can write the equation of the function as: $f(x)=[x-(1+2i)][x-(1-2i)](x-i)(x-(-i))\\=(x-1-2i)(x-1+2i)(x-i)(x+i)\\=x^4-2x^3+6x^2-2x+5$
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