Question

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

Answer 1

`(x-1)` is not a factor of `3x^3-x-3`

Explanation:

In general, if `f(x)` is a polynomial, then `(x-a)` is a factor if and only if `f(a) = 0`.

So in our example, we find:

`f(color(blue)(1)) = 3(color(blue)(1)^3)-color(blue)(1)-3 = 3-1-3 = -1 != 0`

So `(x-1)` is not a factor of `3x^3-x-3`

`color(white)()`
Footnote

If you want to quickly check whether `x=1` is a zero and therefore `(x-1)` 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 `(x+1)` a factor by reversing the signs of the coefficients of the terms of odd degree before adding them.