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To solve the given problem, we need to construct a polynomial \( f(x) \) of degree 3 that has the specified zeros: \( -2 \) with multiplicity 2, and \( 4 \). The solution should be left in factored form.

### Step-by-Step Solution:
1. **Understand the Zeros**:
   - A zero of \( -2 \) with multiplicity 2 means \( (x + 2)^2 \) is a factor of the polynomial.
   - A zero of \( 4 \) means \( (x - 4) \) is also a factor.

2. **Construct the Polynomial**:
   Combining these factors, the polynomial can be written as:
   \[
   f(x) = (x + 2)^2 (x - 4)
   \]

3. **Expand if Necessary**:
   If expansion is required later, you can calculate:
   \[
   (x + 2)^2 = x^2 + 4x + 4
   \]
   Then multiply it by \( (x - 4) \):
   \[
   f(x) = (x^2 + 4x + 4)(x - 4)
   = x^3 - 4x^2 + 4x^2 - 16x + 4x - 16
   = x^3 - 16x - 16
   \]

However, in **factored form**, the answer remains:
\[
f(x) = (x + 2)^2 (x - 4)
\]

Let me know if you'd like further details! Here are some related questions to deepen understanding:

1. How do we handle polynomials with more than one zero of the same value?
2. Why does multiplicity affect the factors of a polynomial?
3. What is the difference between expanded and factored forms of a polynomial?
4. How can we confirm that the given zeros satisfy the polynomial?
5. How does the degree of a polynomial relate to the number of its roots?

**Tip**: Always double-check the multiplicity of zeros to ensure the polynomial matches the given conditions!

Find a polynomial f(x) of degree 3 that has the following zeros: -2 (multiplicity 2), 4. Leave your answer in factored form.

Learn how to find a degree 3 polynomial with zeros -2 (multiplicity 2) and 4, expressed in factored form. This problem involves understanding polynomial functions and the role of multiplicity.

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