# Complex polynomials

• March 18th 2012, 08:00 PM
Skillzz
Complex polynomials
Show that if the polynomial p(z) = anzn+an-1zn-1+…+a0 is written in factored form as p(z) = an(z-z1)d1(z-z2)d2…(z-zr)dr, then
(a) n = d1+d2+…+dr
(b) an-1 = -an(d1z1+d2z2+…drzr)
(c) a0 = an(-1)nz1d1z2d2…zrdr
• March 18th 2012, 08:45 PM
Kiwi_Dave
Re: Complex polynomials
If you multiplied out:

$p(z) = a_n(z-z_1)^{d1}(z-z_2)^{d2}…..(z-z_r)^{dr}$

then how would you form each coefficient so as to make the left hand side look like the right hand side? Note:

$(z-z_1)^{d1}=z^{d1}-d_1.z^{d1-1}z_1^{1}+...+d_1.z^{1}z_1^{d1-1}+(-z_1)^{d1}$

Viewed in this way, you are looking at a problem in combinations and permutations.