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Math Help - Complex numbers-proving and finding roots

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    Complex numbers-proving and finding roots

    Prove that 1+i is a root of the equation z^4+3z^2-6z+10=0. Find all the other roots.
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    Quote Originally Posted by kingkaisai2
    Prove that 1+i is a root of the equation z^4+3z^2-6z+10=0. Find all the other roots.
    z^4+3z^2-6z+10=0

    (1+i)^4+3(1+i)^2-6(1+i)+10

    (1^4+4*1^3*i+6*1^2*i^2+4*1*i^3+i^4) +3(1^2+2*1*i+i^2)-6(1+i)+10

    (1 + 4i - 6 - 4i + 1)+ 3(1 + 2i - 1) - 6(1 + i)  + 10

    -4 + 6i - 6 - 6i + 10

    0 (Check.)

    Now, if 1+i is a root, so is 1-i. Thus we know that \left ( z - (1+i) \right ) \left ( z - (1-i) \right ) = \left ( (z-1) - i \right ) \left ( (z-1) + i \right )  = (z-1)^2 - i^2 = z^2 - 2z + 1 + 1 = z^2 - 2z + 2
    is a factor of z^4+3z^2-6z+10. So do the division. This gives you a quadratic in z to solve, which you can do by factoring or the quadratic formula or whatnot.

    -Dan
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