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.