Final answer:
The quadratic function with a zero of 3i and rational coefficients is found by multiplying the factors (x - 3i)(x + 3i), leading to the equation P(x) = x^2 + 9, so the correct answer is D) P(x) = x^2 + 9.
Step-by-step explanation:
The quadratic function P with a leading coefficient of 1 and a zero of 3i must have rational coefficients, which means it needs a conjugate pair of zeros to ensure the coefficients remain rational. Thus, the other zero must be -3i. We can find the equation by multiplying the factors (x - 3i)(x + 3i), which represents the zero at 3i and its conjugate -3i.
When multiplied, the equation is P(x) = x^2 - (3i)^2. Simplifying this, we get P(x) = x^2 + 9, because - (3i)^2 equals - (3^2 * i^2) which simplifies to - (9 * -1) and gives us +9. The correct choice from the given options is therefore D) P(x) = x^2 + 9.