Thread: 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

Complex roots come in conjugate pairs. Thus, since is a root, so is . Thus and are factors. So the polynomial is divisible by .

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 , you can perform synthetic division twice with and .

Since all the coefficients of your polynomial are real, you know that complex roots occur as conjugates.

Since is a factor, so is , which means is a factor.

So divide your polynomial by to get the remaining quadratic factor, which you can then factorise further into two linear factors.

Hello iFuuZe,

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