Chapter 4.3, Problem 30E

To determine

To Find: Thebinomial factors of each polynomial.

Answer to Problem 30E

The factor of the polynomial is $\left(x-1\right),\left(x+7\right)$ and $\left(x-7\right)$ .

Explanation of Solution

Given:

The given polynomial is $x^3-x^2-49x+49$ .

Calculation:

Consider the function by synthetic division is shown in Table 1

Table 1

$r$	$1$	$-1$	$-49$	$49$
$-7$	$1$	$-8$	$7$	$0$
$-2$	$1$	$-3$	$-43$	135
$-1$	$1$	$-2$	$-47$	$96$
$0$	1	$-1$	$-49$	49
1	1	0	$-49$	$-5$
2	1	1	$-47$	$-5$
7	1	$-8$	7	0

The above table shows that $\left(x-1\right),\left(x+7\right)$ and $\left(x-7\right)$ is the factor.

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

  $\begin{array}{c}f\left(x\right)=x^3-x^2-49x+49\\ f\left(1\right)={\left(1\right)}^3-{\left(1\right)}^2-49\left(1\right)+49\\ f\left(1\right)=0\end{array}$

Then, find the depressed polynomial as,

  $\begin{array}{l}\overline{)1}1-1-4949\\ \underset{\_}{10-49}\\ 10-490\end{array}$

Thus, the value of the depressed polynomial is $x^2-49$ .

Then, factor the depressed polynomial as,

  $\begin{array}{c}{x}^2-49=x^2-7^2\\ =\left(x-7\right)\left(x+7\right)\end{array}$

This shows that the factor of the polynomial is $\left(x-1\right),\left(x+7\right)$ and $\left(x-7\right)$ .