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A friend of mine is asking me for some help to study for an Algebra test. We came across questions in which the problem asked to factor a cubic polynomial. She was only taught the synthetic division method to solve these questions, so it's just down to guess and check to figure out the roots. I know that on her test,

  • the roots will all be integers
  • the coefficient of the cubed term will be always be 1
  • all of the polynomials will be factorable

I was able to point out that you only want to test positive and negative factors of the constant term (by Vieta's). However, some of the problems still take longer than they should. Are there any patterns or simple methods (such as identifying what possible roots should be guessed first or not be guessed at all) one can use to factor such polynomials which could save time?

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    $\begingroup$ By Descartes' rule of signs you can cometimes exclude positive (or negative) roots, which would halve the search field. Also, you can look for modulo patterns across the coefficients to exclude certain divisors, for example $x^3 - q \,x + a\,p^3=0$ with $p,q$ coprime cannot have $p^2$ as a root because $2$ of the $3$ terms would be divisible by $p^3$ but not the third. $\endgroup$ – dxiv Dec 1 '16 at 1:04
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    $\begingroup$ A version of Descartes' rule of signs is that if the signs in the synthetic division don't change for a positive number, you can exclude all larger possibilities (I think that I'm remembering this rule correctly). $\endgroup$ – Michael Burr Dec 1 '16 at 1:18
  • $\begingroup$ Yes thank you I remember that one too now... but does it only apply for positive numbers? $\endgroup$ – Rajat Mittal Dec 1 '16 at 1:29

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