Chapter 4.3, Problem 34E

To determine

To Find: Thebinomial factors of each polynomial.

Answer to Problem 34E

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

Explanation of Solution

Given:

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

Calculation:

Consider the function by synthetic division is shown in Table 1

Table 1

$r$	1	3	3	1
-3	1	0	3	-8
-2	1	1	1	-1
-1	1	2	1	0
0	1	3	3	1
1	1	4	7	8
2	1	5	13	27
3	1	6	21	64

The above table shows that $x+1$ are the factor.

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

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

Thus, the factor is $x+1$ .

Then, find the depressed polynomial as,

  $\begin{array}{l}\overline{)-1}1331\\ \underset{\_}{-10-1}\\ 1210\end{array}$

Thus, the value of the depressed polynomial is $x^2+2x+1$ .

Then, factor the depressed polynomial as,

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

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