Final answer:
The polynomial f(x) given the zeros 7, -1, and -3 is found by multiplying the factors (x - 7), (x + 1), and (x + 3) to get the expanded form f(x) = x^3 - 3x^2 - 25x - 21.
Step-by-step explanation:
We are asked to determine the polynomial f(x) in expanded form given the zeros 7, -1, and -3. By the Factor Theorem, we know if a number, say a, is a zero of a polynomial, then (x - a) is a factor of that polynomial. Therefore, given the zeros, we can write the factors of f(x) as (x - 7), (x + 1), and (x + 3).
Now, we multiply these factors: f(x) = (x - 7)(x + 1)(x + 3)
To expand f(x), we first multiply the factors (x + 1) and (x + 3) to get: x^2 + 4x + 3
Then, we multiply (x - 7) to the result to obtain the final expanded form: f(x) = (x - 7)(x^2 + 4x + 3) = x^3 - 7x^2 + 4x^2 - 28x + 3x - 21
Simplify by combining like terms to get: f(x) = x^3 - 3x^2 - 25x - 21
To ensure the answer is reasonable, compare the factored and expanded forms, and observe that the coefficient in front of the highest degree term (which is 1 in this case) agrees with the product of the leading coefficients of the factors (all of which are 1).