Final answer:
A polynomial function with zeros at x = 5, 2/3, and -7 can be written as f(x) = (x - 5)(x - 2/3)(x + 7).
Step-by-step explanation:
A polynomial function with zeros at x = 5, 2/3, and -7 can be written in factored form as f(x) = (x - 5)(x - 2/3)(x + 7).
This function has zeros when each of the factors is equal to zero. So, when x = 5, the first factor (x - 5) becomes zero; when x = 2/3, the second factor (x - 2/3) becomes zero; and when x = -7, the third factor (x + 7) becomes zero.
To find the polynomial function with these zeros, you multiply the factors together: f(x) = (x - 5)(x - 2/3)(x + 7) = x^3 + (2/3 - 5 + 7)x^2 + (-10/3 + 35/3 - 35/3)x + 5 * 2/3 * -7 = x^3 + (8/3)x^2 + (10/3)x - 70/3.