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I have a cubic equation: $$-x^3 + 7x^2 - 16x + 12 = 0.$$

How they showed us to solve this quickly is to simplify the equation to $$-(x - 2)^2 (x - 3) = 0$$ and find the solutions this way.

My question is how to get to that form in a generic way.

I looked over at certain methods to get the solutions but they all seem to be taking too long and complicated.

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Could you show me on a concrete example so I can accept an answer? – Tool May 7 '12 at 3:01
It's basically systematic trial and error. You know what the divisors of $12$ are (including negative ones); you try them out one by one until you hit the jackpot. – J. M. May 7 '12 at 3:12
up vote 3 down vote accepted

In classes, cubic equations always have rational roots. This is because Cardano's method is such a mess. In your example, all the rational roots are among $\pm(1,2,3,4,6,12)$ so try them. If you find one, you have a quadratic-no problem. Outside class, a numeric solution will be acceptable and root finding will solve the problem, but you should first check the rational roots. If there is one, it may show you something about the problem.

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What if I get this equation: x^3 - 6x^2 - 35x - 38 = 0? It has 2 root solutions. Is there a way to "go easy" with this equation aswell? One of our tests in the past had this. – Tool May 7 '12 at 4:32
There is. If there are rational roots, they are integers that divide $38$. Not too many candidates. Let's cross our fingers and try small ones first. We find that $x=-2$ works. Divide our polynomial by $x+2$. We get $x^2-8x-19$. Use the Quadratic Formula for the other roots. – André Nicolas May 7 '12 at 4:45

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