## Calculus (3rd Edition)

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