Question

Write a polynomial function of minimum degree with real coefficients whose zeros include those listed. Write the polynomial in standard form.

4, -8, and 2 + 5i

Answer 1

is it clear that the first part must include \[(x-4)(x+8)\]?

Answer 2

Use the conjugate root theorem.
For a polynomial of degree n, with real coefficients, if x = z is a root of the polynomial, then the conjugate of z must also be a root of the polynomial.

Thus 2 - 5i must be a root as well.
This means we have 4, -8, 2- 5i and 2 + 5i as roots giving a quartic.

then write out as factors. For the factors with 2 - 5i and 2 + 5i, expand out
(x - (2-5i))(x - (2+5i)) to remove the imaginary parts and come up with a single quadratic factor