How do you use the factor theorem to determine whether x-2 is a factor of P(x)=2x^3 -7x^2 + 7x - 2?

1 Answer
Dec 10, 2015

Since $P \left(2\right) = 0$
$\textcolor{w h i t e}{\text{XXX}} \left(x - 2\right)$ is a factor of $P \left(x\right)$

Explanation:

The Factor Theorem tells us that
$\textcolor{w h i t e}{\text{XXX}} \left(x - a\right)$ is a factor of $P \left(x\right)$ if and only if $P \left(a\right) = 0$

Given
$\textcolor{w h i t e}{\text{XXX}} P \left(x\right) = 2 {x}^{3} - 7 {x}^{2} + 7 x - 2$

then
$\textcolor{w h i t e}{\text{XXX}} P \left(2\right) = 2 \left({2}^{3}\right) - 7 \left({2}^{2}\right) + 7 \left(2\right) - 2$

$\textcolor{w h i t e}{\text{XXX}} = 2 \left(8\right) - 7 \left(4\right) + 7 \left(2\right) - 2$

$\textcolor{w h i t e}{\text{XXX}} = 16 - 28 + 14 - 2$

$\textcolor{w h i t e}{\text{XXX}} = 0$

So $\left(x - 2\right)$ is a factor of $P \left(x\right) = 2 {x}^{3} - 7 {x}^{2} + 7 x - 2$