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Solving a fifth degree polynomial

asked 2016-08-12 12:44:34 -0500

Romuald_314 gravatar image

I want to solve a fifth degree polynomial such as this: x^5-5x^4-10x^3-10x^2-5x-1 == 0

I can't obtain any value, neither the exact value with radicals, nor approximation (with the solve command or the .roots(x) command).

How can I do ? Thx

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answered 2016-08-12 14:36:55 -0500

nbruin gravatar image

Be specific on what kind of roots you want to find (rational ones? real ones? complex ones?)

sage: f=x^5-5*x^4-10*x^3-10*x^2-5*x-1
sage: f.roots(ring=RR)
[(6.72502395887258, 1)]
sage: f.roots(ring=CC)
[(6.72502395887258, 1),
 (-0.461764344593279 - 0.161600091968187*I, 1),
 (-0.461764344593279 + 0.161600091968187*I, 1),
 (-0.400747634843009 - 0.678737070411573*I, 1),
 (-0.400747634843009 + 0.678737070411573*I, 1)]
sage: f.roots(ring=QQbar)
[(6.725023958872576?, 1),
 (-0.4617643445932788? - 0.1616000919681873?*I, 1),
 (-0.4617643445932788? + 0.1616000919681873?*I, 1),
 (-0.4007476348430091? - 0.6787370704115728?*I, 1),
 (-0.4007476348430091? + 0.6787370704115728?*I, 1)]
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All ones. Can I obtain the ones with the exact value (nth roots) instaid of approximations ? Thanks for your answer, it already helps me.

Romuald_314 gravatar imageRomuald_314 ( 2016-08-18 05:02:48 -0500 )edit

Well, if you use ring=QQbar you get objects that are essentially carrying enough information to consider them "exact". However, for roots of fifth degree polynomials, there is likely no better "exact" description of them than "these are roots of this fifth degree polynomial". You can easily get arbitrarily good complex approximations out of elements of QQbar, though.

nbruin gravatar imagenbruin ( 2016-08-19 23:28:25 -0500 )edit

Ok, because I didn't give this example at random, actually it has only one real root explainable with roots (I had read it in a document on polynomials on the web but I can't find it back, I wish I could show you). This is that root I wanted to find...

Romuald_314 gravatar imageRomuald_314 ( 2016-08-25 02:45:53 -0500 )edit

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Asked: 2016-08-12 12:44:34 -0500

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Last updated: Aug 12 '16