Final answer:
The factors of the polynomial are (x + 5), x, (x - 1), and (x - 3).
By multiplying these factors together and simplifying, we get the polynomial function f(x) = x^5 - 4x^4 + 10x^3 - 20x^2 + 15x.
Step-by-step explanation:
To find a polynomial function with the given real zeros, we can start by using the factored form of a polynomial.
The zeros of the function are provided as -5, 0, 1, and 3.
Therefore, the factors of the polynomial are (x + 5), x, (x - 1), and (x - 3).
Since the degree of the polynomial is 4, we need to multiply these factors to find the polynomial function.
(x + 5)(x)(x - 1)(x - 3) = x(x + 5)(x - 1)(x - 3)
Next, we can expand this expression to get the polynomial function.
Let's simplify step by step:
x(x + 5)(x - 1)(x - 3) = x(x^2 - x + 5x - 5)(x - 3)
= x(x^3 - x^2 + 5x - 5)(x - 3)
= (x^4 - x^3 + 5x^2 - 5x)(x - 3)
= x^5 - 4x^4 + 10x^3 - 20x^2 + 15x
Therefore, the polynomial function with the given real zeros and degree 4 is f(x) = x^5 - 4x^4 + 10x^3 - 20x^2 + 15x.