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

Jan 19, 2017

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

#### Explanation:

In general, if $f \left(x\right)$ is a polynomial, then $\left(x - a\right)$ is a factor if and only if $f \left(a\right) = 0$.

So in our example, we find:

$f \left(\textcolor{b l u e}{1}\right) = 3 \left({\textcolor{b l u e}{1}}^{3}\right) - \textcolor{b l u e}{1} - 3 = 3 - 1 - 3 = - 1 \ne 0$

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

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Footnote

If you want to quickly check whether $x = 1$ is a zero and therefore $\left(x - 1\right)$ is a factor of a polynomial, just add the coefficients and see if the result is $0$.

This is the same as evaluating the polynomial for $x = 1$ since $1$ raised to any integer power is $1$.

You can also check whether $x = - 1$ is a zero and $\left(x + 1\right)$ a factor by reversing the signs of the coefficients of the terms of odd degree before adding them.