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What is the simplest method to solve an equation of 3rd degree.

For example: $$-x^{3} + x^{2} + x - 1 = 0$$

Please I don't want the resolution of this equation I just want the simplest method to use to solve it, then I'll try to solve it on my own.

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marked as duplicate by Ross Millikan, copper.hat, Amzoti, Arkamis, Davide Giraudo Apr 29 '13 at 17:30

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multiply both sides by -1 and try grouping method –  imranfat Apr 29 '13 at 16:58

2 Answers 2

up vote 1 down vote accepted

$-x^{3} + x^{2} + x - 1 = 0$

$-x^{2}(x-1)+x-1=0$

$(x-1)(1-x^{2})=0$

$x=1$ and $x^{2}=1$

$x=1 , x=-1$

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He (or she) actually didn't want the answer, just the method ... –  Dolma Apr 29 '13 at 17:04
    
for 3 degree always not exist a algebra manner some time you have to use numerical analysis for a close answer as famous algoritms in numerical analysis –  Somaye Apr 29 '13 at 17:07
    
Yes you're absolutely right. However here it's not the case, and the OP seemed like he wanted to find the roots on his own and just wanted some hints to get going ;) –  Dolma Apr 29 '13 at 17:10
    
somaye's comment is completely untrue. There are closed form solutions for degrees 3 and 4, it only stops being the case after degree 5. –  Ulysse Mizrahi Apr 29 '13 at 17:14
    
I like your method it's simple than the native one, but how did you pass from the second line to the third ? –  Aimad Majdou Apr 29 '13 at 17:20

A possible method (if there's an easy first root to find : usually an integer not so far from $0$)

  1. Find one root $r$ for the polynomial
  2. Factor the polynomial by $(x-r)$
  3. Solve the resulting 2nd degree polynomial
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