Write a third degree polynomial function with rational coefficients in standard form with the given zeros: 4, 3i

Answer 1

For a polynomial equation with rational coefficients, any complex solutions (with nonzero imaginary part) will always come in conjugate pairs; thus the existence of the root 3i implies that -3i will also be a root. With this in mind, we can construct the factored form of the desired cubic equation:

(x - 4)(x - 3i)(x + 3i) = 0.

now we multiply the factors together and simplify to reach standard form:

(x - 4)(x² + 9) = 0

x³ - 4x² + 9x - 36 = 0