what is the fastest way of factorising a cubic equation

Question

For example i need to factorise the equation $x^3-6x^2+11x-6=0$ I know the method of putting values in the equation and then check for which value the equation becomes zero (here for x=2 the equation is zero) then I divide the equation by x-2 by long division method and I get the quotient but the process is lengthy .

isn't there any other way for saving time?

Answer 1

you can depend on $(x-1)^3=x^3-3x^2+3x-1$ so $$x^3-6x^2+11x-6=x^3-3x^2-3x^2+3x+8x-1-5$$ $$(x-1)^3-3x^2+8x-5$$ $$(x-1)^3-(x-1)(3x-5)$$

Answer 2

Standard exercise on high school identities: \begin{align*}x^3-6x^2+11x-6&= (x-2)^3-12x+8+11x-6=(x-2)^3-x+2\\&=(x-2)\bigl((x-2)^2-1\bigr)=(x-2)(x-3)(x-1).\end{align*}