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.

To start, we can simplify the function by factoring out the common factor 4.
f(x)=4x^3-4x^2-16x+16=x^3-x^2-4x+4.

We could divide f(x) by (x-2) and solve the resulting quadratic equation.

We could, however, use another property of f(x).

We are given that (x-2) is a factor

Knowing that the constant term is +4, and using the factor theorem, we can try (x+2) as a factor.
f(-2)=-8-4+8+4=0(x+2) is a factor

We note that f(x) has a sum of coefficients 1-1-4+4=0, equal to zero, this means that f(1)=0, or (x-1) is a factor(x-1)

Therefore all the factors of the function are (x+2)(x-2)(x-1)
and the roots are x={1,2,-2}