Question

How do you use the remainder theorem and the factor theorem to determine whether (b+4) is a factor of `(b^3 + 3b^2 − b + 12)`?

Answer 1

`(b+4)` is a factor of `b^3+3b^2-b+12`

Explanation:

Remainder theorem states that if we divide a polynomial function `f(x)` by `(x-a)`, the remainder is `f(a)`.

Hence, if `(x-a)` is a factor of `f(x)`, `f(a)=0` (This is factor theorem.)

As we have to determine whether `(b+4)` is a factor of `f(b)=b^3+3b^2-b+12` or not,

we should evaluate `f(-4)` which is equal to

`(-4)^3+3(-4)^2-(-4)+12=-64+48+4+12=0`

Hence `(b+4)` is a factor of `b^3+3b^2-b+12`.