Question

find all the other zeroes of the polynomial x^4+4x^3-4x^2-32x-32 if two of its zeroes are 2 root 2 and -2 root 2

Answer 1

If x = 2√2 and x = -2√2
then two of the factors are
(x - 2√2) and (x + 2√2)
or (x^2 - 8) must be a factor of your polynomial.

Do a long algebraic division to get

x^4+4x^3-4x^2-32x-32
= (x^2 - 8)(x^2 - 4x + 4)

we can get more roots from
x^2 - 4x + 4 = 0
(x-2)(x-2) = 0 ---> a perfect square
so x = 2

so the roots are 2 , ± 2√2

Answer 2

Thank you