Final answer:
A polynomial function with real zeros at -1, 1, and 3, and a degree of 3, is obtained by multiplying the factors (x + 1), (x - 1), and (x - 3). The resulting polynomial is f(x) = x^3 - 3x^2 - x + 3.
Step-by-step explanation:
To form a polynomial function with given real zeros at -1, 1, and 3, and a degree of 3, we need to set up factors that represent these zeros. A zero of a polynomial is a value of x for which the polynomial evaluates to zero. According to the Factor Theorem, if 'a' is a zero of the polynomial f(x), then (x - a) is a factor of f(x).
Given the zeros -1, 1, and 3, our polynomial function will have factors corresponding to each zero: (x + 1), (x - 1), and (x - 3). To obtain the polynomial function, we simply multiply these factors together:
f(x) = (x + 1)(x - 1)(x - 3)
Expanding this, we get:
f(x) = (x2 - 1)(x - 3)
Further expanding gives us the final polynomial:
f(x) = x3 - 3x2 - x + 3
Thus, the desired polynomial function is f(x) = x3 - 3x2 - x + 3.