Question

write a polynomial function P(x) with rational coefficients so that P(x)=0 has the given roots -6,3, and -15i

Answer 1

First, suppose that the given roots were {-6,3}.
The related factors would be (x-[-6]) and (x+3).
Multiplying these together results in (x+6)(x-3) = x^2 +3x -18.

Note that all of the coefficients of this polynomial are real and rational.

If there's an imaginary root, such as -15i, then the related factor is (x-[-15i]), or (x+15i). You could multiply the previous quadratic polynomial obtained above by this new factor. Unfortunately, the resulting 3rd order poly would not have rational coeff.

Any idea how to get around that?

Answer 2

Here's what I'd try (note the word, "try"): I'd assume that there is one more root, and that that root is the conjugate of -15i, which would be 15i.

Then your two imaginary factors would be (x+15i) and (x-15i).

Multiply out all four factors (2 that involve real roots and 2 that involve imaginary roots). That will give you a polynomial with rational coefficients.