I need help finding coefficients of polynomial.

Question

I have the polynomial $p(z)=az^{2}+bz+c$ .

I also have the roots:

$z_0=-6$

$z_1=6-i$

And I also know that:

$p(-i)=-6$

Given this information I have to find the coefficients $a,b, c$.

I know that a polynomial with roots $z_0$ and $z_1$ will have the factors $(z-z_0)(z-z_1)$ so what I've tried is:

But this approach does not take into consideration that $p(-i)=-6$ and is therefore the wrong solution.

Answer 1

Your approach is fine; you've forgotten two things.

1. It has those as factors, but could have OTHER factors as well

2. The other factors can only be a constant, because you know it's a quadratic and has only two roots. So you just need to say

$$ A(z-z_0)(z-z_1) = A(z+6)(z-6+i) = \ldots $$

and then you need to use your final constraint to determine the value of $A$.