Question

How do you use the factor theorem to determine whether x+3 is a factor of `2x^3 + x^2 – 13x + 6`?

Answer 1

For `f(x)=2x^3+x^2-13x+6`
If `(x+3)` is a factor, `x=-3` should be a root.
So for `x=-3` to be a root, `f(-3)=0`
If `f(-3)!=0`, `(x+3)` is not a factor.

Explanation:

Factor theorem basically boils down to testing potential roots in the equation.

So for some equation `f(x)`

A root is an `x` value which satisfies `f(x)=0`

So if a factor of `f(x)` is `(x-a)`,
`x=a` is a root of `f(x)`.

So for your case,
`f(x)=2x^3+x^2-13x+6`,
we want to see if `(x+3)` is a factor.
So we test `x=-3`

`f(-3)=-54+9+39+6=0`
`f(-3)=0` so `x=-3` is a root.
Therefore `(x+3)` is indeed a factor.