https://file.aiquickdraw.com/mathbot/file/1730736011264ymra5xt5.jpg

Let's analyze the question and each option carefully.

The function given is:
\[
f(x) = x^3 + 4x^2 + 2x - 1
\]

We are told that \((-1, 0)\) is a zero of the function, meaning \(x = -1\) is a root. This is a cubic polynomial, so it can have up to three roots (real or complex).

Let’s evaluate each of the answer choices:

1. **"There are three more zeros."**  
   This would imply there are four zeros in total, which is incorrect because a cubic polynomial can have at most three zeros.

2. **"All the zeros must be real."**  
   Knowing one real zero does not necessarily mean all zeros must be real. A cubic polynomial can have one real root and two complex roots.

3. **"There are two more zeros."**  
   This is likely correct. Since we already know one root \((-1)\), and a cubic polynomial has exactly three roots (counting multiplicities), there must be two additional roots.

4. **"All the zeros must be irrational."**  
   This is not necessarily true. A polynomial with rational coefficients, like this one, can have rational or irrational roots depending on the nature of its factors. We cannot conclude irrationality without further factoring or using the Rational Root Theorem.

### Conclusion:
The correct answer is:
- **"There are two more zeros."**

Would you like a detailed breakdown of polynomial roots or further clarification?

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Here are five related questions that expand on this topic:

1. How many roots can a polynomial of degree \(n\) have in general?
2. Can a polynomial of degree 3 have all complex roots?
3. How can we determine if the remaining roots are real or complex?
4. What methods can be used to factorize cubic polynomials?
5. How does the Rational Root Theorem help in finding roots of polynomials?

**Tip:** For polynomials with real coefficients, if you have one real root, any remaining complex roots must occur in conjugate pairs.

If (-1, 0) is a zero of the polynomial function f(x) = x^3 + 4x^2 + 2x - 1, what can you say about the remaining zeros of this function without factoring?

Explore the zeros of the polynomial function f(x) = x^3 + 4x^2 + 2x - 1, given that (-1, 0) is a root. This problem demonstrates the Fundamental Theorem of Algebra and involves analyzing possible real and complex roots without factoring. Ideal for high school students in Grades 10-12.

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