How do you find an nth degree polynomial function with real coefficients if you know its zeros and its leading coefficient?

To find an nth degree polynomial function with given zeros and a leading coefficient of 1, follow these steps:

1. Identify Zeros: You are given that the zeros of the polynomial are 2 and 3i. Since the polynomial must have real coefficients, it is important to note that complex zeros come in conjugate pairs. Therefore, if 3i is a zero, then -3i must also be a zero.
2. Form Factors: The factors of the polynomial based on the zeros are: (x - 2) for the real zero 2, and (x - 3i) and (x + 3i) for the complex zeros. This gives us the factors:
• (x - 2)
• (x - 3i)
• (x + 3i)
3. Write the Polynomial: The polynomial can be expressed as the product of its factors:

P(x) = (x - 2)(x - 3i)(x + 3i)

4. Multiply the Complex Factors: Multiply the complex factors first:

(x - 3i)(x + 3i) = x^2 + 9 (this uses the difference of squares).

5. Combine All Factors: Now, substitute this back into the polynomial:

P(x) = (x - 2)(x^2 + 9).

6. Expand the Polynomial: Now, expand this expression:

P(x) = x(x^2 + 9) - 2(x^2 + 9)

7. Combine Like Terms: This gives:

P(x) = x^3 + 9x - 2x^2 - 18.

8. Final Form: Rearranging gives:

P(x) = x^3 - 2x^2 + 9x - 18.

Therefore, the expanded and simplified polynomial of degree 3 with real coefficients that satisfies the given conditions is:

P(x) = x^3 – 2x^2 + 9x – 18