Chapter 4.6, Problem 59WE

To determine

To Find:

To find the other factor of $x^3-3x^2+3x-2$ if one of factor is $\left(x-2\right)$ .

Answer to Problem 59WE

The other factor of the polynomial $x^3-3x^2+3x-2$ is $\left(x^2-x+1\right)$ .

Explanation of Solution

Given information:

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

Formula:

Rational root theorem:

  $\pm \frac{dividers{\text{ of a}}_0}{dividers{\text{ of a}}_n}$ where $a_0,a_n$ -integers.

To Factorize $x^3-3x^2+3x-2$ ,

Apply rational root theorem $\pm \frac{dividers{\text{ of a}}_0}{dividers{\text{ of a}}_n}$ ,

Here $a_0=2;a_n=1$ ,

The dividers of $a_0-1,2$ and the dividers of $a_n=1$

Therefore, check the following rational numbers $\pm \frac{1,2}{1}$ ,

  $\frac{2}{1}$ is a root of the expression so factor out $\left(x-2\right)$ .

  $\Rightarrow \left(x-2\right)\frac{x^3-3x^2+3x-2}{\left(x-2\right)}$

Simplify,

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

Therefore, other factor of the polynomial $x^3-3x^2+3x-2$ is $\left(x^2-x+1\right)$ .