# Solving a fifth degree polynomial

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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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)]

more

All ones. Can I obtain the ones with the exact value (nth roots) instaid of approximations ? Thanks for your answer, it already helps me.

( 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.

( 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...

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