
Complex polynomials
Show that if the polynomial p(z) = a_{n}z^{n}+a_{n1}z^{n1}+…+a_{0} is written in factored form as p(z) = a_{n}(zz_{1})^{d1}(zz_{2})^{d2}…(zz_{r})^{dr}, then
(a) n = d_{1}+d_{2}+…+d_{r}
(b) a_{n1} = a_{n}(d_{1}z_{1}+d_{2}z_{2}+…d_{r}z_{r})
(c) a_{0} = a_{n}(1)^{n}z_{1}^{d1}z_{2}^{d2}…z_{r}^{dr}

Re: Complex polynomials
If you multiplied out:
$\displaystyle p(z) = a_n(zz_1)^{d1}(zz_2)^{d2}…..(zz_r)^{dr}$
then how would you form each coefficient so as to make the left hand side look like the right hand side? Note:
$\displaystyle (zz_1)^{d1}=z^{d1}d_1.z^{d11}z_1^{1}+...+d_1.z^{1}z_1^{d11}+(z_1)^{d1}$
Viewed in this way, you are looking at a problem in combinations and permutations.