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Math Help - Complex polynomials

  1. #1
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    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)nz1d1z2d2zrdr
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  2. #2
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    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.
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