# How do you determine whether x-1 is a factor of the polynomial x^4-3x^3+2x^2-x+1?

Jan 31, 2017

$\left(x - 1\right) \text{ is a factor of } p \left(x\right)$.

#### Explanation:

Let us denote the given polynomial by $p \left(x\right)$, so that,

$p \left(x\right) = {x}^{4} - 3 {x}^{3} + 2 {x}^{2} - x + 1.$

To determine whether $\left(a x + b\right)$ is a factor of $p \left(x\right)$, we check
$p \left(- \frac{b}{a}\right)$. If this is $0 , \left(a x + b\right)$ is a factor of $p \left(x\right) .$

So, for $\left(x - 1\right) ,$ we have to check $p \left(1\right)$.

$p \left(1\right) = 1 - 3 + 2 - 1 + 1 = 4 - 4 = 0$

$\Rightarrow \left(x - 1\right) \text{ is a factor of } p \left(x\right)$.

It will be clear from the above discussion that, to determine whether

$\left(x - 1\right)$ is a factor of a given poly., we have to simply check whether

the sum of the co-effs. is $0.$