Question

Use the Factor Theorem to prove x^3 - 13x - 12 is divisible by x^2 – X – 12

Answer 1

See Below.

Explanation:

We want to prove that:

`x^3-13x-12\text{ is divisible by } x^2-x-12`

We can factor the divisor:

`x^2-x-12=(x-4)(x+3)`

According to the Factor Theorem, if we have a polynomial P(x) divided by a binomial in the form of (x - a) and if P(a) = 0, then the binomial is a factor of P(x).

Our two binomial factors our (x - 4) and (x + 3). Thus, a = 4 and a = -3.

Evaluate the polynomial for both of these factors:

`P(4)=(4)^3-13(4)-12=0`

And:

`P(-3)=(-3)^3-13(-3)-12=0`

Since both yielded zero, the original polynomial is divisible by both (x - 4) and (x + 3) or x² - x - 12. Hence:

`x^3-13x-12\text{ is indeed divisible by } x^2-x-12`