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 2 - Linear and Quadratic Functions - Section 2.3 Quadratic Functions and Their Zeros - 2.3 Assess Your Understanding - Page 146: 64


Zeros: $-1,2$ $x$-intercepts: $-1,2$

Work Step by Step

To find the zeros of a function $f$, solve the equation $f(x)=0$ The zeros of the function are also the $x-$intercepts. Let $g(x)=0$: $$x^6-7x^3-8=0$$ Let $u=x^3$, the original equation becomes $$u^2-7u-8=0$$ By factoring $$(u+1)(u-8) = 0$$ Use the Zero-Product Property by equating each factor to zero, then solve each equation to obtain: \begin{align*} u+1 &= 0 &\text{ or }& &u-8=0\\ u &= -1 &\text{ or }& &u=8\\ \end{align*} To solve for $x$, we use $u=x^3$ $$\because u = x^3$$ $$\therefore x = u^{\frac{1}{3}}$$ For $u=-1$ $$x=(-1)^{\frac{1}{3}} $$ $$x= -1$$ For $u=8$ $$x=(8)^{\frac{1}{3}}$$ $$x= 2$$ $\therefore x = -1,2$
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