Question

How do you use the factor theorem to determine whether x+4 is a factor of `2x^3 + x^2 - 25x + 12`?

Answer 1

Since when `x=-4` then `2x^3+x^2-25x+12=0`
by the factor theorem `(x+4)` is a factor.

Explanation:

The factor theorem says that for an expression `f(x)`
then `(x-a)` is a factor of `f(x)` if and only if `f(a)=0`

Note that `(x+4) = (x- color(red)((-4)))`

So we evaluate
`color(white)("XXX")f(color(red)(-4)) = 2(-4)^3+-4)^2-25(-4)+12`

`color(white)("XXXXXXX")=-128 +16 +100 +12`

`color(white)("XXXXXXX")=0`

and discover that `(x-(-4)) = (x+4)` is a factor of the given expression.