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To solve this problem, we need to express the polynomial \( f(x) = x^3 + 8 \) in factored form using its complex zeros. Here’s the step-by-step solution:

### Step 1: Recognize the Complex Zeros
The complex zeros of \( f(x) = x^3 + 8 \) are given as:
\[
-2, \quad 1 + i\sqrt{3}, \quad 1 - i\sqrt{3}
\]
These are the roots of the polynomial.

### Step 2: Write \( f(x) \) in Factored Form
Since we know the roots, we can write \( f(x) \) as the product of terms of the form \( (x - \text{root}) \) for each root:
\[
f(x) = (x + 2)(x - (1 + i\sqrt{3}))(x - (1 - i\sqrt{3}))
\]

### Step 3: Simplify the Complex Conjugate Terms
Notice that \( (x - (1 + i\sqrt{3}))(x - (1 - i\sqrt{3})) \) involves a pair of complex conjugates, which we can simplify as follows:
\[
(x - (1 + i\sqrt{3}))(x - (1 - i\sqrt{3})) = \left( x - 1 \right)^2 - \left( i\sqrt{3} \right)^2
\]
Since \( i^2 = -1 \), we get:
\[
= (x - 1)^2 - (-3) = (x - 1)^2 + 3
\]
Expanding \( (x - 1)^2 + 3 \):
\[
= x^2 - 2x + 1 + 3 = x^2 - 2x + 4
\]

### Step 4: Write the Final Factored Form
Now we can express \( f(x) \) completely in factored form:
\[
f(x) = (x + 2)(x^2 - 2x + 4)
\]

This is the complete factorization of \( f(x) = x^3 + 8 \) using its complex zeros.

Would you like more details or have any questions?

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Here are five related questions to expand on this problem:
1. How do you factor other polynomials with real and complex roots?
2. What is the significance of complex conjugate pairs in polynomial factorization?
3. How do you derive the complex roots of a cubic equation like \( x^3 + 8 = 0 \)?
4. Why is it useful to express a polynomial in its factored form?
5. How does one determine if a polynomial can be factored into linear factors over complex numbers?

**Tip:** When dealing with polynomials that have complex roots, always look for complex conjugate pairs to simplify the factorization process.

Find the complex zeros of the polynomial function f(x) = x^3 + 8 and write f in factored form.

Learn how to find and use the complex zeros of the polynomial f(x) = x^3 + 8 to write it in factored form. This solution covers essential algebraic concepts and uses the Complex Conjugate Root Theorem. Suitable for students in Grades 10-12.

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