Trying to express a polynomial as $(z-\text{root}_1)(z-\text{root}_2)$

Question

I'm probably making some stupid mistake, but here's my problem: I have the polynomial $0.5z^2+9iz-0.5$, the roots I calculated are $a=-9i-4\sqrt5i,\;\;\;b=-9i+4\sqrt5i$.

I tried to express the polynomial as $(z-a)(z-b)$, and I plugged some numbers to check if it works - but it seems like I always get double the result, what am I doing wrong?

(Results were calculated with a calculator)

Answer 1

Your polynomial is equal to $$0.5(z-a)(z-b)$$

You forgot to multiply by the coefficient of $z^2$. The factored form of a quadratic function $az^2+bz+c$ is $$a(z-z_1)(z-z_2)$$ where $z_1,z_2$ are roots of the polynomial.

Answer 2

$$0.5z^2+9iz-0.5=0.5(z^2+18iz-1)=0.5[z-(-9i-4\sqrt5i)][z-(-9i+4\sqrt5i)]$$