Factor the polynomial f(x)=x4−4x3+16x−16 given that 2 and −2 are zeros. Factored form: f(x)=

Question

then identify all real zeros

Answer 1

Remember that a is a zero of f(x) if and only if (x-a) is a factor of f(x).

So because 2 is a zero, (x-2) is a factor of f(x).

Also -2 is a zero, so (x+2) is a factor of f(x).

Since they are both factors then (x-2)(x+2)=x^2-4 is a factor of f(x). Use polynomial long division to then factor it out.

You will get that x^4-4x^3+16x-16 = (x^2-4)(x^2-4x+4)

We can then factor both of the quadratics nicely x^2-4=(x-2)(x+2) and x^2-4x+4 = (x-2)(x-2).

So f(x) = (x-2)(x+2)(x-2)(x-2) = (x-2)^3(x+2)

Thus your answers are the factored form of f(x)=(x-2)^3(x+2).

The real zeros of f(x) are 2 and -2.