College Algebra 7th Edition

Published by Brooks Cole
ISBN 10: 1305115546
ISBN 13: 978-1-30511-554-5

Chapter 3, Polynomial and Rational Functions - Section 3.5 - Complex Zeros and the Fundamental Theorem of Algebra - 3.5 Exercises - Page 329: 11


(a) $\left\{-i, i \right\}$ (b) $P(x) = (x-i)^2(x+i)^2$

Work Step by Step

$\bf{(a) \text{ Zeros}}$ Factor the polynomial completely to obtain: $P(x) = (x^2+1)(x^2+1) \\P(x) = [(x-i)(x+i)] \cdot [(x-i)(x+i)]$ Equate each unique factor to zero then solve each equation to obtain: \begin{array}{ccc} &x-i=0 &\text{or} &x+i=0 \\&x=i &\text{or} &x=-i\end{array} Thus, the zeros of the function are: $\left\{-i, i \right\}$. $\bf{(b) \text{ Completely Factored Form}}$ From part (a) above, the completely factored form of $P(x)$ is: $P(x) = (x-i)(x+i)(x-i)(x+i) \\P(x) = (x-i)^2(x+i)^2$
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