Question

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

Answer 1

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

Explanation:

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

`1^3-3(1)^2+4(1)-2 = 1 - 3 + 4 - 2 = 0`

Thus `x-1` is a factor of `x^3-3x^2+4x-2`.

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

`x^3-3x^2+4x-2 = (x-1)(x^2-2x+2)`