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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?

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    $\begingroup$ The rational zero theorem is the way to go $\endgroup$ – imranfat Sep 6 '16 at 15:45
  • $\begingroup$ I really don’t think that the method you describe is at all time-consuming. $\endgroup$ – Lubin Sep 6 '16 at 15:50
  • $\begingroup$ @Lubin time consuming since I have to keep checking the output for many values $\endgroup$ – danny Sep 6 '16 at 15:54
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you can depend on $(x-1)^3=x^3-3x^2+3x-1$ so $$x^3-6x^2+11x-6=x^3-3x^2-3x^2+3x+8x-1-5$$ $$(x-1)^3-3x^2+8x-5$$ $$(x-1)^3-(x-1)(3x-5)$$

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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*}

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