Trouble with factoring polonomial to the 3rd degree

Question

I am having trouble factoring this problem: $\displaystyle{-x^{3} + 6x^{2} - 11x + 6}$

I know the answer but i can't figure out how it is done with this. I have tried by grouping and is doesn't seem to work. Can someone show me how to do this.

Answer 1

$$\begin{align} & -x^3+6x^2-11x+6 \\ =& -x^3+1+6x^2-11x+5 \\ =& (1-x^3)+6x^2-6x-5x+5 \\ =& (1-x)(1+x+x^2)-6x(1-x)+5(1-x) \\ =& (1-x)(1+x+x^2-6x+5) \\ =& (1-x)(x^2-5x+6) \\ =& (1-x)(x^2-2x-3x+6) \\ =& (1-x)(x(x-2)-3(x-2)) \\ =& (1-x)(x-2)(x-3) \end{align}$$

Answer 2

Hint: $$-1 + 6 - 11 + 6 = 0$$

This method may be suggested by examining the rational roots theorem, as well.

Answer 3

Big hint:

In this case, just look for zeros of the cubic polynomial, since each zero tells you a linear factor (by the factor theorem).

Then try looking at factors of 6 (the constant term) - there are not many integer factors of 6 ...