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To express \( f(x) = x^3 - 3x^2 + 9x + 13 \) as a product of linear factors, let's go through the following steps, using the information that \( -1 \) is a zero of \( f(x) \).

### Steps:
1. **Divide the polynomial by \( x + 1 \)** since \( -1 \) is a root.
2. This division will yield a quotient (a quadratic polynomial) and potentially a remainder of zero.
3. Factor the resulting quadratic expression if possible.

Let's proceed.

#### Step 1: Synthetic Division
To divide \( f(x) = x^3 - 3x^2 + 9x + 13 \) by \( x + 1 \), we use synthetic division with \( -1 \) as the divisor.

- Coefficients of \( f(x) \): \( 1, -3, 9, 13 \)
- Synthetic division with \( -1 \):
  - Bring down the 1.
  - \( 1 \times -1 = -1 \), add to -3 to get -4.
  - \( -4 \times -1 = 4 \), add to 9 to get 13.
  - \( 13 \times -1 = -13 \), add to 13 to get 0 (no remainder).

So, the quotient from this division is:
\[
x^2 - 4x + 13
\]

#### Step 2: Factor the Quotient
Now, we have:
\[
f(x) = (x + 1)(x^2 - 4x + 13)
\]

The quadratic \( x^2 - 4x + 13 \) does not factor further over the real numbers, but we can factor it over the complex numbers using the quadratic formula:
\[
x = \frac{4 \pm \sqrt{16 - 52}}{2} = \frac{4 \pm \sqrt{-36}}{2} = 2 \pm 3i
\]

So, we can express \( x^2 - 4x + 13 \) as:
\[
(x - (2 + 3i))(x - (2 - 3i))
\]

### Final Answer:
Thus, the polynomial \( f(x) \) in terms of linear factors is:
\[
f(x) = (x + 1)(x - (2 + 3i))(x - (2 - 3i))
\]

Would you like further details on any of these steps? 

Here are 5 additional questions to deepen your understanding:
1. How does synthetic division work, and why is it useful for factoring polynomials?
2. Can you identify cases when a quadratic polynomial can’t be factored over real numbers?
3. How does the quadratic formula help in finding complex roots?
4. Why do polynomials with complex roots always come in conjugate pairs?
5. How can you verify if a given factorization is correct?

**Tip:** When given a polynomial root, use synthetic division to simplify the polynomial degree, making it easier to factor further.

For the polynomial below, -1 is a zero.

f(x) = x^3 - 3x^2 + 9x + 13

Express f(x) as a product of linear factors.

Learn how to express the polynomial f(x) = x^3 - 3x^2 + 9x + 13 as a product of linear factors using synthetic division and the quadratic formula. This solution demonstrates factoring with complex numbers and provides a step-by-step breakdown for students in Grades 11-12.

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