This is for future reference because it took me forever to remember how to do it the other day.
The problem is to find an nth degree polynomial that satisfies a specific set of conditions:
- n = 3
- 2 and 4i are zeros
- f(2) = -51
First and foremost, since this is a 3rd degree polynomial we need three roots, but are given only two. The third one comes from the fact that imaginary roots come in conjugate pairs. This means that since 4i is a root -4i must also be. Now we have three roots.
Now, those three roots can be used to determine three linear factors of the polynomial in the form (x - a) along with an unknown factor, a:
f(x) = a(x - 2)(x - 4i)(x + 4i)
f(x) = a(x - 2)(x - 4i)(x + 4i)
