Fin all roots exactly (rational, irrational and imaginary) for the polynomial equation

-1 is a root by rational root theorem.

-1| 1 0 -4 -4 -1

-1 1 3 1

1 -1 -3 -1 0

Using synthetic division we see that (x

^{4}+0x^{3}-4x^{2}-4x-1)/(x-(-1))=x^{3}-x^{2}-3x-1Again -1 is a root by the rational roots theorem

Again we use synthetic division

-1| 1 -1 -3 -1

-1 2 1

1 -2 -1 0

(x

^{3}-x^{2}-3x-1)/(x+1)=x^{2}-2x-1 whose roots are, by the quadratic formula(2±√8)/2 or 1+√2 and 1-√2