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We are given the polynomial zeros \( x = 5 \), \( x = \frac{2}{3} \), and \( x = -7 \). To find the correct polynomial function from the choices, we need to write the corresponding factors.

### Step 1: Convert Zeros to Factors
- If \( x = 5 \) is a zero, then \( (x - 5) \) is a factor.
- If \( x = \frac{2}{3} \) is a zero, we rewrite this as \( 3x = 2 \), or \( (3x - 2) \) is a factor.
- If \( x = -7 \) is a zero, then \( (x + 7) \) is a factor.

Thus, the polynomial function will have factors:
\[
f(x) = (x - 5)(3x - 2)(x + 7)
\]

### Step 2: Compare with the Given Options
- **Option A**: \( f(x) = (x + 5)(2x + 3)(x - 7) \) (This does not match our required factors.)
- **Option B**: \( f(x) = (x + 5)(3x + 2)(x - 7) \) (Again, wrong signs and factors.)
- **Option C**: \( f(x) = (x - 5)(2x - 3)(x + 7) \) (Has the correct signs for two factors but wrong for the middle factor.)
- **Option D**: \( f(x) = (x - 5)(3x - 2)(x + 7) \) (This matches our desired polynomial.)

Thus, the correct answer is **Option D**.

Would you like more details or have any questions?

Here are five related questions to expand on this topic:
1. How do you determine the factor from a fractional root?
2. What is the relationship between the degree of a polynomial and its zeros?
3. How do you expand a factored polynomial to its standard form?
4. Can a polynomial have complex roots, and how are those factored?
5. What is the difference between a root and a zero of a polynomial?

**Tip:** The number of zeros (real or complex) of a polynomial always equals the degree of the polynomial.

Which polynomial function has zeros when x = 5, 2/3, -7?

Learn how to find the polynomial function with zeros at x = 5, 2/3, and -7. This problem covers key concepts such as factoring, the Zero Product Property, and polynomial functions, suitable for high school algebra students.

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