Precalculus (6th Edition) Blitzer

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

Chapter 2 - Section 2.5 - Zeros of Polynomial Functions - Exercise Set - Page 379: 66


See the explanation below.

Work Step by Step

Suppose that f(x) represents a polynomial equation of degree n. According to the Fundamental Theorem of Algebra, the equation has at least one complex root; we’ll call it ${{c}_{1}}$. According to the Factor Theorem, we know that $\left( x-{{c}_{1}} \right)$ is a factor of f(x). Equating equation (1) equal to 0, we get $f\left( x \right)={{a}_{n}}\left( x-{{c}_{1}} \right)\left( x-{{c}_{2}} \right).....\left( x-{{c}_{n}} \right)=0$ Then, ${{a}_{n}}\left( x-{{c}_{1}} \right)=0$ This implies that either ${{a}_{n}}=0$ or $(x-{{c}_{1}})=0$. So, the order of the given polynomial $f\left( x \right)$ in the equation (1) has to be at least 1. Similarly, \[\] \[\begin{align} & \left( x-{{c}_{2}} \right)=0 \\ & \left( x-{{c}_{3}} \right)=0 \\ & . \\ & . \\ & . \\ & \left( x-{{c}_{n}} \right)=0 \\ \end{align}\] Hence, the roots of the given polynomial in equation (1) can be ${{c}_{2}},{{c}_{3}},......,{{c}_{n}}$. Thus, for a polynomial of $n^{th}$ order, there are n roots.
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