How can I find the root of this polynomial?

Question

How can I find the root of this polynomial?

$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

Answer 1

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 $60x^4$?

Answer 2

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.