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To find all zeros of the function \( f(x) = x^3 + 9x^2 + 31x + 39 \), we need to find the values of \( x \) for which \( f(x) = 0 \).

### Steps to Solve:

1. **Check Rational Roots**: Use the Rational Root Theorem to test possible rational roots. The potential rational roots are given by \( \pm \) factors of the constant term (39) divided by the factors of the leading coefficient (1).

2. **Factorize the Polynomial**: If any rational roots are found, factor them out using synthetic division or polynomial division.

3. **Solve the Resulting Quadratic**: After factoring out the rational root, solve the resulting quadratic equation.

Let's go through these steps systematically.The zeros of the function \( f(x) = x^3 + 9x^2 + 31x + 39 \) are:

1. \( x = -3 \) (real root)
2. \( x = -3 - 2i \) (complex root)
3. \( x = -3 + 2i \) (complex root)

These are the exact values of the zeros. Do you need a detailed explanation of any steps or further assistance?

### Related Questions
1. How do you use the Rational Root Theorem to find potential rational roots?
2. What is synthetic division, and how is it used to factor polynomials?
3. How do you solve a quadratic equation with complex coefficients?
4. What does it mean for a function to have complex roots?
5. How can you graphically interpret the zeros of a polynomial function?

**Tip:** For complex roots, they always occur in conjugate pairs if the coefficients of the polynomial are real numbers.

Consider the function f(x) = x^3 + 9x^2 + 31x + 39. List all zeros of the function. Enter exact values, separated by commas as necessary.

Learn how to find the zeros of the cubic polynomial function f(x) = x^3 + 9x^2 + 31x + 39 using the Rational Root Theorem and factoring methods. This problem involves complex numbers and polynomials, making it suitable for students in Grades 10-12.

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