# How do you use the factor theorem to determine whether a-1 is a factor of a^3 – 2a^2 + a – 2?

Mar 10, 2018

$\left(a - 1\right)$ is not a factor of ${a}^{3} - 2 {a}^{2} + a - 2$

#### Explanation:

According to factor theorem if $x - a$ is a factor of polynomial function $f \left(x\right)$, then $f \left(a\right) = 0$. Now let $\left(x - 1\right)$ be a factor of the function

$f \left(x\right) = {a}_{0} {x}^{n} + {a}_{1} {x}^{n - 1} + {a}_{2} {x}^{n - 2} + \ldots \ldots . + {a}_{n}$

then $f \left(1\right) = {a}_{0} + {a}_{1} + {a}_{2} + \ldots \ldots . + {a}_{n} = 0$

This means that if $\left(a - 1\right)$ is a factor of polynomial function ${a}^{3} - 2 {a}^{2} + a - 2$, then sum of its coefficients must be zero.

Here as $1 - 2 + 1 - 2 = - 2$ and sum of coefficients is not zero,

$\left(a - 1\right)$ is not a factor of ${a}^{3} - 2 {a}^{2} + a - 2$