Chapter 4.3, Problem 36E

To determine

To Find: Thenumber of times $-1$ is the root of the $x^3+2x^2-x-2=0$ .

Answer to Problem 36E

The factor r $-1$ is the factor of the polynomial only one time.

Explanation of Solution

Consider the given equation is,

  $x^3+2x^2-x-2=0$

Consider the synthetic division for the given polynomial as,

  $\begin{array}{l}\overline{)-}12-1-2\\ \underset{\_}{-1-12}\\ 11-20\end{array}$

Since, the remainder is zero, therefore $-1$ is the root of the given equation.

Factorize the depressed polynomial as,

  $\begin{array}{l}{x}^2+x-2=\left(x+2\right)\left(x-1\right)\\ x=-2,1\end{array}$

Thus, the factor r $-1$ is the factor of the polynomial only one time.