Chapter 4.3, Problem 32E

To determine

To Find: Thebinomial factors of each polynomial.

Answer to Problem 32E

The factor of the polynomial is $\left(x+2\right)\left(x-2\right)$ and $\left(x-2\right)$ .

Explanation of Solution

Given:

The given polynomial is $x^3-2x^2-4x+8$ .

Calculation:

Consider the function by synthetic division is shown in Table 1

Table 1

$r$	$1$	-2	-4	8
-3	1	-5	11	-25
-2	1	-4	4	0
-1	1	-3	-1	9
0	1	-2	-4	8
1	1	-1	-5	3
2	1	0	-4	0
3	1	1	-1	5

The above table shows that $\left(x+2\right)\left(x-2\right)$ are the factor.

Then, test some values by the help of the factor theorem as,

  $\begin{array}{c}f\left(x\right)=x^3-2x^2-4x+8\\ f\left(1\right)={\left(1\right)}^3-2{\left(1\right)}^2-4\left(1\right)+8\\ f\left(1\right)=3\end{array}$

Also,

  $\begin{array}{c}f\left(x\right)=x^3-2x^2-4x+8\\ f\left(2\right)={\left(2\right)}^3-2{\left(2\right)}^2-4\left(2\right)+8\\ f\left(2\right)=0\end{array}$

Thus, the factor is $x-2$ .

Then, find the depressed polynomial as,

  $\begin{array}{l}\overline{)2}1-2-48\\ \underset{\_}{20-8}\\ 10-40\end{array}$

Thus, the value of the depressed polynomial is $x^2-4$ .

Then, factor the depressed polynomial as,

  $x^2-4=\left(x+2\right)\left(x-2\right)$

This shows that the factor of the polynomial is $\left(x+2\right)\left(x-2\right)$ and $\left(x-2\right)$ .