Chapter 5.6, Problem 24PPS

To determine

To calculate: the remaining factors of the polynomial.

Answer to Problem 24PPS

The remaining factors are $\left(x+1\right)\left(x-1\right)$

Explanation of Solution

Given information:

  $x+2$ is a factor of the polynomial $x^3+2x^2-x-2$

Formula used:

Synthetic Substitution :- Synthetic division is used to find the value of a function.

Factor Theorem :- The binomial $x-r$ is a factor of the polynomial $P(x)$ if and only if $P(r)=0$ .

Remainder Theorem:- If a polynomial $P(x)$ is divided by $x-r$ , the remainder s a constant $P(r)$ , and

  $P(x)=Q(x)(x-r)+P(r)$ ,

Where $Q(x)$ is a polynomial with degree one less than $P(x)$ .

Calculation:

Consider ,

  $x^3+2x^2-x-2$

Since, the binomial $x+2$ is a factor of the polynomial by using Synthetic division we get,

  $\begin{array}{l}\overline{)-2}\begin{array}{cccc}1& 2& -1& -2\end{array}\\ \begin{array}{ccc}-2& 0& 2\end{array}\\ \overline{\begin{array}{ccc}1& 0& -1\end{array}\hspace{.2em}|}\begin{array}{c}\text{ 0}\end{array}\end{array}$

Thus, the remainder is zero. The polynomial $x^2-1$ can be factored as $(x+2)(x^2-8)$ .

The polynomial $x^2-1$ is the depressed polynomial.

The remaining factors of the polynomial are :-

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

Hence, the factors of the polynomial are $\left(x+2\right)\left(x+1\right)\left(x-1\right)$ .