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Let's solve this question using synthetic division to determine if \( i \), \( -i \), and \( -2 \) are zeros of the polynomial \( f(x) = x^3 + 2x^2 + x + 2 \).

### Steps for Synthetic Division:
1. **First test \( i \):**
   The coefficients of the polynomial are \( [1, 2, 1, 2] \) corresponding to \( x^3, x^2, x, \) and the constant term. We divide this by \( x - i \).

2. **Second test \( -i \):**
   If \( i \) isn't a zero, we'll check the same set of coefficients with synthetic division using \( x + i \).

3. **Third test \( -2 \):**
   For \( -2 \), we perform synthetic division to determine if it is a zero.

I’ll now work through the synthetic division calculations for each value.The synthetic division shows that:

- \( i \) is a zero of the polynomial \( f(x) \).
- \( -i \) is a zero of the polynomial \( f(x) \).
- \( -2 \) is also a zero of the polynomial \( f(x) \).

Thus, all three values are zeros of the polynomial. The correct answer is:

**Yes, Yes, Yes**.

Would you like further details or explanations? Here are some related questions to consider:

1. How does synthetic division work step-by-step for complex numbers?
2. Why does the polynomial have complex zeros like \( i \) and \( -i \)?
3. What is the relationship between a polynomial's degree and its number of real and complex zeros?
4. How would the graph of \( f(x) = x^3 + 2x^2 + x + 2 \) look based on its zeros?
5. How can you factor a cubic polynomial using its zeros?

**Tip:** Always check all possible zeros of a polynomial, including complex and negative values, when determining the full set of solutions.

Using synthetic division, determine whether the numbers i, -i, -2 are zeros of the polynomial f(x) = x^3 + 2x^2 + x + 2.

Learn how to use synthetic division to check if i, -i, and -2 are zeros of the cubic polynomial f(x) = x^3 + 2x^2 + x + 2. This problem explores polynomial division, zeros, and the application of complex numbers. Suitable for high school students.

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