find the zeros of the following polynomial f(x)=x^4-2x^3-5x^2+8x+4

Question

i need help to find the zeros, thanks in advance

Answer 1

First find the rational zeroes. List these: -1, -2, -4, 1, 2, 4. Some of these may be actual zeroes.

We substitute each of these into the polynomial until we get two actual zeroes. It turns out that x=-2 and 2 make f(x)=0. 16+16-20-16+4=0, 16-16-20+16+4=0.

Use synthetic division to divide by -2:

-2 | 1 -2 -5  8  4

      1 -2  8 -6 -4

      1 -4  3  2  0.

Then divide this result by 2:

2 | 1 -4  3  2

     1  2 -4 -2

     1 -2 -1  0 = x²-2x-1, x²-2x+1-2=0, (x-1)²=2, x=1±√2.

Zeroes are -2, 2, 1+√2, 1-√2.