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: 43

Answer

(a) $ (x^2+\sqrt 2x+1)(x^2-\sqrt 2x+1)$ (b) $\frac{-\sqrt 2\pm i\sqrt {2}}{2},\frac{\sqrt 2\pm i\sqrt {2}}{2}$.

Work Step by Step

(a) $f(x)=x^4+1=x^4+2x^2+1-2x^2=(x^2+1)^2-2x^2=(x^2+\sqrt 2x+1)(x^2-\sqrt 2x+1)$ (b) For $x^2+\sqrt 2x+1=0$, we have $x=\frac{-\sqrt 2\pm\sqrt {2-4}}{2}=\frac{-\sqrt 2\pm i\sqrt {2}}{2}$. For $x^2-\sqrt 2x+1=0$, we have $x=\frac{\sqrt 2\pm\sqrt {2-4}}{2}=\frac{\sqrt 2\pm i\sqrt {2}}{2}$.
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