How Do You Construct a Degree 8 Polynomial with Given Complex and Real Roots?

[Caldus]

Sorry to keep asking questions...my teacher is horrible...

Anyway, I need to find a polynomial equation with degree 8 and contains the roots (3 - 2i), (-5 + 7i), 0 (with multiplicity 3), -3, and 4 (with multiplicity 2).

This is what I came up with:

p(x) = (x - (3 - 2i))(x - (-5 + 7i))(x)^3(x + 3)(x - 4)^2

Two things I'm wondering about (besides whether I got it right of course):

1. Don't I need to add in the conjugates for 3 - 2i and -5 + 7i in the equation? In that case, the degree would have to be 10, no?

2. Can I simplify this any further?

Thanks again.

 

[Hurkyl]

You only need to include the complex conjugates if you are looking for a real polynomial; they're not necessary for complex polynomials.