How to find the roots of this 4th order polynomial?

Question

Can someone explain how to factor/find roots to this 4th order polynomial:

$$ s^4 + 14s^3 +45s^2 +650s + 1800 = 0 $$

It's such a nightmare. I've been stuck for hours, any help would be appreciated :)

Answer 1

Those roots can be found algebraically (e.g. see here), but the procedure is laboursome and error-prone, so that some numerical method (Newton-Raphson, fixed point iteration, secant procedure, ..) might be more practical.

If you manage to guess roots you can reduce the order of the polynomial and end up with easier algebraic solution formulae.

Answer 2

$s^4 + 14s^3 +45s^2 +650s + 1800$ is irreducible over $\mathbb Q$ because it is irreducible mod $11$.

In particular, its roots are irrational numbers.

There is a formula for the roots but it'll probably be very ugly.