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 3.2 - Polynomial Functions and Their Graphs - 3.2 Exercises - Page 302: 44


$P(x)=(x-1)^2(x^2+x+1)^2$ Zero: $1$ with multiplicity $2$ Refer to the graph below.

Work Step by Step

Factor the polynomial completely to obtain: \begin{align*} P(x)&=(x^3-1)(x^3-1)\\ &=(x-1)(x^2+x+1)(x-1)(x^2+x+1)\\ &=(x-1)^2(x^2+x+1)^2 \end{align*} To find the zeros, use the Zero-Product Property by equating each factor to $0$, then solve each equation to obtain: \begin{align*} (x-1)^2&=0 &\text{or}& &(x^2+x+1)^2=0\\ x-1&=\pm\sqrt0 &\text{or}& &x^2+x+1=\pm\sqrt0\\ x-1&=0 &\text{or}& &x^2+x+1=0\\ x&=1 \text{ (multiplicity 2)}&\text{or}& &\text{ no real solution}\\ \\ \end{align*} The zero of the function is $1$ with multiplicity $2$. Use a graphing utility to graph $P(x)$. Refer to the graph above.
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