I need help to factorise the following polynomial:
$x^4 – 2x^3 + 8x^2 – 14x + 7$
The solution I need to reach is $(x-1)(x^3 – x^2 + 7x – 7)$. I need to factorize to this exactly as it is for a limit question where I cancel out the $(x-1)$ in the numerator and denominator. How do I proceed?
Best Answer
A brute force approach is to notice that
$$\begin{align} x^4-2x^3+8x^2-14x+7 =&(x^4-x^3)-(x^3-x^2)+7(x^2-2x+1)\\ =&x^3(x-1)-x^2(x-1)+7(x-1)^2\\ =&(x-1)(x^3-x^2+7x-7)\\ =&(x-1)\left(x^2(x-1)+7(x-1)\right)\\ =&(x-1)(x-1)(x^2+7) \end{align}$$
QED