Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 2 - Section 2.3 - Polynomial Functions and Their Graphs - Exercise Set - Page 349: 29

Answer

The zeros are $x=0$ with multiplicity 1 and $x=1$ with multiplicity 2. At $x=0$, the graph will cross the $x\text{-axis}$ and at $x=1$, the graph will touch the $x\text{-axis}$ and turn around.

Work Step by Step

For zeros, let $f\left( x \right)=0$. That is, $\begin{align} & {{x}^{3}}-2{{x}^{2}}+x=0 \\ & x\left( {{x}^{2}}-2x+1 \right)=0 \\ & x{{\left( x-1 \right)}^{2}}=0. \end{align}$ Then, the values of x are as follows: $x=0,\ 1$. For multiplicity, $x=0$, with a multiplicity 1, because of $x$. $x=1$, with a multiplicity 1, because of ${{\left( x-5 \right)}^{2}}$. The multiplicity of $x=0$ is odd; the graph will cross the $x\text{-axis}$. The multiplicity of $x=1$ is even; the graph will touch the $x\text{-axis}$ and turn around.
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