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.
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.)