Find a polynomial equation with real coefficients that has the given roots.
4 and -3i
Since the given roots are 4 and -3i, the conjugate of -3i is 3i. Therefore, the polynomial equation with these roots can be formed by multiplying the factors (x - 4)(x + 3i)(x - 3i):
(x - 4)(x + 3i)(x - 3i)
= (x - 4)(x^2 - (3i)^2)
= (x - 4)(x^2 + 9)
= x^3 + 9x - 4x^2 - 36
Thus, the polynomial equation with real coefficients and the given roots is:
p(x) = x^3 - 4x^2 + 9x - 36