One factor of f(x)=4x3-4x2-16x+16 is (x – 2). what are all the roots of the function? use the remainder theorem.

Remark

For the first two terms, factor out 4x^2

For the second two terms, factor out 16

The factor using the difference of squares.

Solution

f(x) = 4x^2(x - 1) - 16(x - 1)       x - 1 is a common factor.

f(x) = (4x^2 - 16) (x - 1)            factor out 4 from the first factor.

f(x) = 4(x^2 - 4)(x - 1)               Factor the difference of squares

x^2 - 4 = (x - 2)(x + 2)             Substitute this difference in the line above

f(x) = 4(x - 2)(x + 2)(x - 1)

Answer

The roots are 2, - 2, 1