# Determining the number of complex roots (including multiplicities) of a polynomial

Could someone please explain/show me how to determine the number of complex roots including multiplicities of a polynomial such as

$P(z):= 5i z^{37} - (6 +2i)z^{4} + 4z^2 - i$

Would i need to factorise it so it is in the form

$P(z) = a_n(z - w_1)(z- w_2)...(z-w_n)$?

I know that every polynomial of degree $n \geq 1$ has preciely $n$ roots in $\mathbb C$..

Otherwise is there a quick and easy way of doing such a thing? Or is there a theorem i could use?

Any help much appreciated. Thank you.

• I don't understand your question. You know that every polynomial has $n$ roots in $\Bbb C$ (this counts multiplicity!), so your polynomial has $37$ roots in $\Bbb C$ counting multiplicity. – user98602 Mar 27 '14 at 17:28
• oh cool, sorry i thought that just seemed to easy and that maybe is wasnt 37.. haha thanks. – Bernard.Mathews Mar 27 '14 at 17:42

As you said, the fundamental theorem of algebra says that every polynomial of degree $n$ with complex coefficients has $n$ complex roots (counting multiplicity). So your polynomial has 37 roots in $\Bbb C$ (again, counting multiplicity.)