Final answer:
A polynomial with zeros at -1, 2, and 7 can be represented by the function f(x) = (x + 1)(x - 2)(x - 7). Multiplying these factors will give the polynomial in its expanded form.
Step-by-step explanation:
If a polynomial function has zeros at x = -1, x = 2, and x = 7, then you can write a function rule for this polynomial by translating the zeros into factors. Since all zeros have a multiplicity of 1, each zero will correspond to a simple linear factor.
For a zero at x = -1, the factor is (x + 1).
For a zero at x = 2, the factor is (x - 2).
For a zero at x = 7, the factor is (x - 7).
By multiplying these factors together, we will create a polynomial that has these zeros. Therefore, a polynomial function rule that represents this function is:
f(x) = (x + 1)(x - 2)(x - 7)
You can leave the function in its factored form or expand it to produce a standard form polynomial. The expanded form will produce a cubic polynomial.