Question

Completely factorize the following polynomial: x⁴−3x³+4x²−8

Answer 1

$$x^4-3x^3+4x^2-8$$

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

$$=x^3(x-2)-x^2(x-2)+2(x-2)(x+2)$$

$$=(x-2)(x^3-x^2+2x+4)$$

$$=(x-2)((x^3+x^2)-(2x^2+2x)+(4x+4))$$

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

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

$$=(x-2)^3(x+1)$$

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