The zeros of a polynomial are the values for which the polynomial equals zero. In this case, we are given the zeros -7 and 3, with -7 having a multiplicity of 1 and 3 having a multiplicity of 2. Multiplicity refers to the number of times a root appears in a polynomial.
First, remember that if a number n is a root of a polynomial, then (x - n) is a factor of that polynomial. Therefore, the zero -7 contributes a factor of (x - (-7)) = (x + 7) to the polynomial.
Next, the root 3 has a multiplicity of 2, so it contributes a factor of (x - 3)^2 to the polynomial.
By multiplying these factors together, we form a polynomial: (x + 7)(x - 3)^2.
The degree of a polynomial is determined by adding up the multiplicities of all roots. In this case, -7 has a multiplicity of 1 and 3 has a multiplicity of 2, so the degree of the polynomial is 1 + 2 = 3, which matches the given degree.
By comparing the formed polynomial to the provided options, you can see that it matches with answer choice A: f(x)=(x+7)(x−3)^2. Therefore, option A is the correct solution to the problem.