# Thread: Complex (non-real) factors of a polynomial

1. ## 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

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

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 $\displaystyle z^2+9$, you can perform synthetic division twice with $\displaystyle 3i$ and $\displaystyle -3i$.

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

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

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

4. Hello iFuuZe,

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