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The problem asks to find all zeros, real and non-real, of the cubic polynomial:

\[
f(x) = x^3 + 9x^2 + 19x - 29
\]

To find the zeros, we can apply methods such as synthetic division, the Rational Root Theorem, or numerical methods to solve the cubic equation.

Let me solve it step by step to find both real and complex roots.The zeros of the cubic polynomial \( f(x) = x^3 + 9x^2 + 19x - 29 \) are:

\[
1, -5 - 2i, -5 + 2i
\]

This includes one real root \(1\) and two complex conjugate roots \( -5 - 2i \) and \( -5 + 2i \).

Would you like further details on how these roots were found, or do you have any other questions?

Here are 5 related questions you might find useful:
1. What is the Rational Root Theorem and how is it applied?
2. How do synthetic division and polynomial division work?
3. How do you factor polynomials with complex roots?
4. Can all cubic equations be solved using the quadratic formula?
5. What is the geometric interpretation of complex roots?

**Tip:** When solving higher-degree polynomials, start by looking for simple rational roots using trial and error with possible divisors of the constant term.

Find all zeros, real and non-real, of f(x) = x^3 + 9x^2 + 19x - 29.

Learn how to find all zeros, both real and non-real, of the cubic polynomial f(x) = x^3 + 9x^2 + 19x - 29. This problem involves algebraic methods such as the Rational Root Theorem, synthetic division, and complex root analysis. Suitable for students in Grades 10-12.

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