Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 3 - Differentiation - 3.3 Product and Quotient Rules - Exercises - Page 123: 65


(a) $c = −1$ is a multiple root (b) $c = −1$ is not a multiple root

Work Step by Step

(a) Consider $$ f(x)=x^{5}+2 x^{4}-4 x^{3}-8 x^{2}-x+2$$ Since $$ f'(x)=5x^{4}+8 x^{3}-12x^{2}-16 x-1$$ Then $$ f(-1)=f'(-1)=0 $$ Hence $c = −1$ is a multiple root of $f(x)$ (b) Given $$f(x) =x^{4}+x^{3}-5 x^{2}-3 x+2 $$ Since $$f'(x) =4x^{3}+3x^{2}-10x-3 $$ Then \begin{align*} f(-1)&= 0\\ f'(-1)&= 6\neq 0 \end{align*} Hence $c = −1$ is not multiple root of $f(x)$
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