How can I find the root of this polynomial?

Question

$f(x) =x^5 - 15x^3-10x^2 +60x -20 $

My attempts : By fundamental theorem of algebra, every odd degree polynomial has at least one root.

But here I don't know how to find the root? And how to break this given polynomial

WA tells us that this polynomial has no nice roots. – lhf Apr 26 '18 at 14:10 — lhf, Apr 26 '18 at 14:10

score 1 · Answer 1 · answered Apr 26 '18 at 14:04

Often it is only possible to find the roots by numerical methods. Sometimes one can guess a root. But there is no formula for a root of a 5th degree polynomial as Galois theory tells us. Are You sure that the polynomial is correctly written? Or must it be $60x^4$?

score 0 · Answer 2 · answered Apr 26 '18 at 14:06

I don't know what you mean by "break this polynomial", but the usual procedure to try first is to see if there are any rational roots, which look like factors of $20$ divided by factors of $1$.

If that doesn't work, you must resort to numerical methods, because your polynomial is a quintic, and there is no general solution to the quintic.

How can I find the root of this polynomial?

2 Answers2