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Math Help - Complex (non-real) factors of a polynomial

  1. #1
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    Complex (non-real) factors of a polynomial

    Hi the question says If z-3i is a factor of 2z^4-4z^3+21z^2-36z+27

    I tried doing long division by dividing it by z-3i but i get a perfect zero as a answer so i cant do much with that, is their another way i can approach this.? and im working in the chapter factorasation of polynomials of C.

    thanks
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  2. #2
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    Complex roots come in conjugate pairs. Thus, since 3i is a root, so is -3i. Thus z-3 and z+3 are factors. So the polynomial is divisible by (z-3i)(z+3i)=z^2+9.

    You can now perform long division to get a quadratic polynomial. You can then find the roots of this quadratic polynomial by completing the square or using the quadratic formula.


    Remark: Instead of doing long division with z^2+9, you can perform synthetic division twice with 3i and -3i.
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  3. #3
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    Since all the coefficients of your polynomial are real, you know that complex roots occur as conjugates.

    Since \displaystyle z - 3i is a factor, so is \displaystyle z + 3i, which means \displaystyle (z - 3i)(z + 3i) = z^2 + 9 is a factor.

    So divide your polynomial by \displaystyle z^2 + 9 to get the remaining quadratic factor, which you can then factorise further into two linear factors.
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  4. #4
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    Hello iFuuZe,

    If you mean that you get zero as the remainder, that's good.
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