Question

Select all zeros of the polynomial function.

g(x)=x5 + 3x4-4x³2x² - 12x - 16
-4
1
-3i
-3
2
-21
iv 2
-2
3
-i√3
iv 3
-1
4
-i√/2
21
0
5
сл
-i
3i

Answer 1

hi,

`Roots\ are\ -1, 2,4,√(2) i,-√(2) i`

Explanation:

`x^5+3x^4-4x^3-2x^2-12x-16\\=x^5+x^4+2x^4+2x^3-6x^3-6x^2+4x^2+4x-16x-16\\=x^4(x+1)+2x^3(x+1)-6x^2(x+1)+4x(x+1)-16(x+1)\\=(x+1)(x^4+2x^3-6x^2+4x-16)\\=(x+1)(x^4-2x^3+4x^3-8x^2+2x^2-4x+8x-16)\\=(x+1)(x^3(x-2)+4x^2(x-2)+2x(x-2)+8(x-2))\\=(x+1)(x-2)(x^3+4x^2+2x+8)\\=(x+1)(x-2)(x^2(x+4)+2(x+4))\\=(x+1)(x-2)(x+4)(x^2+2)\\\\=(x+1)(x-2)(x-4)(x+√(2) i)(x-√(2)i)\\ \\Roots\ are\ -1, 2,4,√(2) i,-√(2) i`