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Thread: complex polynom

  1. #1
    Junior Member
    May 2009

    complex polynom

    hello, i have to determine the complex zero values of the function z^5+2z^3+z

    ok here my try:

    z (z^4+2z^2+1) = 0 => gives first zero pt. at 0
    now substitution z^4 =u^2 and z^2 = u

    now with pq formula i get this: -1+- 0

    but what now ??
    thanks in advance !
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  2. #2
    Junior Member nimon's Avatar
    Sep 2009
    Edinburgh, UK

    Repeated Roots

    Let z^{5}+2z^{3}+z = p(z).

    By the remainder theorem you know that:
    a is a root of p(z) \Leftrightarrow p(z)=(z-a)q(z) for some poloynomial q(z).
    You did this correctly for a=0 to get
    p(z) = (z-0)q(z) = z(z^{4}+2z^{2}+1).
    Then by your substitution you found that u=z^{2}=-1, hence z=(-1)^{\frac{1}{2}}=\pm i, so that we get
    p(z)=z(z^{2}+1)(z^{2}+1) = z(z+i)(z-i)(z+i)(z-i).
    Therefore the roots are 1,i,-i. The reason that you got 3 distinct roots instead of the 5 you might expect with an order 5 polynomial is that the factors (z+i) and (z-i) are both repeated in the (unique) factorisation of p(z). So really there are 5 roots, it's just that i and -i are repeated.
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