Question

Synthetic division And the Remainder theorem

Answer 1

The remainder theorem says that if $x-c$ is a factor of a polynomial $p(x)$, then the remainder upon dividing $\dfrac{p(x)}{x-c}$ is 0 so that there exists a lower degree polynomial $q(x)$ as the quotient:

$\dfrac{p(x)}{x-c}=q(x)$

Using the fact that $x-2$ is a factor, you can find a quadratic $q(x)$ which is easy to factorize further.

Synthetic division yields

$q(x)=4x^2+4x-8$

which can be factored further as

$4(x^2+x-2)=4(x+2)(x-1)$

So,

$f(x)=4x^3-4x^2-16x+16=4(x-2)(x+2)(x-1)$

The roots are then $x=-2,1,2$.

Answer 2

Answer:

The answer to this question is "B.) x = -2, x = 1, or x = 2"