Final answer:
The zeros of the function f(x) = (x+5)^2(x-8)^3(x+7)^2 are -7, -5, and 8, with the zero at -5 and -7 each having multiplicity 2, and the zero at 8 having multiplicity 3.
Step-by-step explanation:
To find the zeros of the function f(x) = (x+5)^2(x-8)^3(x+7)^2, we need to set the function equal to zero and solve for the values of x that make the equation true. The zeros are the x-intercepts of the graph of the function, which occur where each factor equals zero.
Setting each factor to zero gives us:
- x + 5 = 0 leads to x = -5
- x - 8 = 0 leads to x = 8
- x + 7 = 0 leads to x = -7
Since the powers of the factors indicated are 2 and 3, these are repeated zeros, which means:
- The zero at x = -5 has multiplicity 2.
- The zero at x = 8 has multiplicity 3.
- The zero at x = -7 has multiplicity 2.
Thus, f(x) has three distinct zeros at x = -7, x = -5, and x = 8, with multiplicities as indicated.