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

Jan 2, 2017

Observe that ${x}^{3} - 3 {x}^{2} + 4 x - 2$ evaluates to $0$ at $x = 1$ and conclude that $x - 1$ is a factor of ${x}^{3} - 3 {x}^{2} + 4 x - 2$.

#### Explanation:

In general, given a polynomial $P \left(x\right)$, we have that $x - a$ is a factor of $P \left(x\right)$ if and only if $P \left(a\right) = 0$. To test if $x - 1$ is a factor of ${x}^{3} - 3 {x}^{2} + 4 x - 2$, then, we can evaluate ${x}^{3} - 3 {x}^{2} + 4 x - 2$ at $1$:

${1}^{3} - 3 {\left(1\right)}^{2} + 4 \left(1\right) - 2 = 1 - 3 + 4 - 2 = 0$

Thus $x - 1$ is a factor of ${x}^{3} - 3 {x}^{2} + 4 x - 2$.

If we want to see how it factors out, we can use polynomial long division to find that

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