Sum and product of a complex polynom I'm looking for the sum and the product of the roots of
$$ z^7 + (3-i)z^6+\pi z^3+e = 0 $$
I think this can be done without calculating the roots (the sum should be -3+i and the product -e) but I don't understand it. How do you solve this? And what's the theory behind it?

If a monic polynomial $P$ has roots $\alpha_1,\cdots,\alpha_n$, then $P(x)=(x-\alpha_1)\cdots(x-\alpha_n)$. Now expand this product and get $P(x)=x^n-(\alpha_1+\cdots+\alpha_n)x^{n-1}+\cdots+(-1)^n(\alpha_1\cdots\alpha_n)$.
If $P$ is not monic, then $\dfrac1{a_n}P$ is monic and has the same roots as $P$. Then the argument above applies to give that the sum of the roots is $-\dfrac{a_{n-1}}{a_n}$ and the product of the roots is $(-1)^n\dfrac{a_0}{a_n}$.
See also Vieta's formulas.
