Question

How do you use the factor theorem to determine whether x-4 is a factor of `x^3-21x+20`?

Answer 1

The factor theorem states that if `f(x_0)=0`, then `(x-x_0)` divides `f(x)`. So, `(x-4)` divides `x^3-21x+20` if and only if the polynomial is zero when `x=4`.

Let's do the computation:

`f(4)= 4^3 - 21*4+20 = 64-84+20 = 0`

So yes, `(x-4)` is a factor of `x^3-21x+20`. Indeed, the three roots are `-5`, `1` and `4`, so we have

`x^3-21x+20=(x-1)(x-4)(x+5)`