Question

Find all possible complex roots of x^4-3x^3-6x^2+8x and of x^3+10x^2+169x

Answer 1

Hello,

1)
$x^{4}-3x^{3}-6x^{2}+8x=x(x^3-3x^2-6x+8)$

$=x(x^{3}-x^{2}-2x^2+2x-8x+8)=x(x^{2}(x-1)-2x(x-1)-8(x-1))$

$=x(x-1)(x^2-2x-8)=x(x-1)(x^2+2x-4x-8)$

$=x(x-1)(x(x+2)-4(x+2))=x(x-1)(x+2)(x-4)$


2)

$x^{3}+10x^{2}+169x=x(x^2+10x+169)=x(x+5-12i)(x+5+12i)$