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 - Chapter Review - Review Exercises - Page 268: 28


remaining zeros: $-i$ and $1-i$ The equation of the function is: $f(x)= x^4-2x^3+3x^2-2x+2$

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 real number coefficients, then its conjugate $(p –i q)$, is also a zero of the function. We notice that the function has a degree of $4$, so it has $4$ complex or real zeros. We are given $2$ complex zeros, so the remaining zeros are also complex (because they come in pairs). Thus, the remaining $2$ zero are: $-i$ and $1-i$, which are the conjugates of $i$ and $1+i$, by the Conjugate Pairs Theorem. We can use the zeros to form factors, which we multiply to find the polynomial: $f(x)=(x-i) [x-(1+i)][x-(-i)][x-(1-i)] \\=(x^2-i^2) [(x-1)^2 -i^2] \\=(x^2+1) (x^2-2x+1+1) \\ = x^4-2x^3+3x^2-2x+2$
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