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Math Help - Factorise P(z)=9z^3+(9i-12)z^2+(5-12i)z+5i over C if P(-i)=0

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    Factorise P(z)=9z^3+(9i-12)z^2+(5-12i)z+5i over C if P(-i)=0

    Factorise over C if P(-i)=0

    For all the book's questions so far we were told to find one factor after finding a root, then use long division to find the next factor and so on. But I am not sure where to start here because of the "i"s in the polynomial.

    So all I know so far is that (Z-(-i)) = (z+i) is a factor. But I don't think I will be able to divide 9z^3+(9i-12)z^2+(5-12i)z+5i by (z+i)

    Thank you
    Attached Thumbnails Attached Thumbnails Factorise P(z)=9z^3+(9i-12)z^2+(5-12i)z+5i  over C if P(-i)=0-wolframalpha-20110205194940385.gif  
    Last edited by anees7112; February 5th 2011 at 05:13 PM.
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    Quote Originally Posted by anees7112 View Post
    Factorise over C if P(-i)=0

    For all the book's questions so far we were told to find one factor after finding a root, then use long division to find the next factor and so on. But I am not sure where to start here because of the "i"s in the polynomial.

    So all I know so far is that (Z-(-i)) = (z+i) is a factor. But I don't think I will be able to divide 9z^3+(9i-12)z^2+(5-12i)z+5i by (z+i)

    Thank you
    You should be able to do the long division - it's no different than if it was \displaystyle z - a with \displaystyle a \in \mathbf{R} as a factor...
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    Ah ok . Thank you very much ! I did not realise I could
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    Quote Originally Posted by anees7112 View Post
    Factorise over C if P(-i)=0

    For all the book's questions so far we were told to find one factor after finding a root, then use long division to find the next factor and so on. But I am not sure where to start here because of the "i"s in the polynomial.

    So all I know so far is that (Z-(-i)) = (z+i) is a factor. But I don't think I will be able to divide 9z^3+(9i-12)z^2+(5-12i)z+5i by (z+i)

    Thank you
    Here is an alternative

    P(z)=9z^3+(9i-12)z^2+(5-12i)z+5i

    P(z)=(z+i)\left(9z^2+kz+5)=9z^3+kz^2+5z+9iz^2+kiz+  5i

    P(z)=9z^3+(9i+k)z^2+(5+ki)z+5i

    Both the coefficient of z^2 and the coefficient of z reveal k.
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