Synthetic division and the remainder theorem

one factor of f(x)=4x^3-4x^2-16x+16 is (x – 2). what are all the roots of the function? use the remainder theorem.
a) x = 1, x = 2, or x = 4
b) x = –2, x = 1, or x = 2
c) x = 2, x = 4, or x = 16
d) x = –16, x = 2, or x = 16

The remainder theorem says that if is a factor of a polynomial , then the remainder upon dividing is 0 so that there exists a lower degree polynomial as the quotient:



Using the fact that is a factor, you can find a quadratic which is easy to factorize further.

Synthetic division yields



which can be factored further as



So,



The roots are then .

It is the top one. the answer is -7. that is smallest