Final answer:
A polynomial f(x) of degree 4 with zeros 3 (multiplicity 2), 0, and -4 is f(x) = x(x - 3)²(x + 4) in factored form.
Step-by-step explanation:
To find a polynomial f(x) of degree 4 with the given zeros 3 (multiplicity 2), 0, and -4, we use the zero product property. The multiplicity of a zero indicates how many times that particular zero is repeated as a root of the polynomial. So with 3 being a root with multiplicity 2, it will appear twice in the factorization of the polynomial.
The zeros of the polynomial give us the factors of the polynomial. The zero 3 with multiplicity 2 gives us the factor (x - 3)², the zero 0 gives us the factor x, and the zero -4 gives us the factor (x + 4).
The polynomial in factored form is therefore: f(x) = x(x - 3)²(x + 4).