Final answer:
To find zeros and factor the polynomials, factors such as x are taken out, and techniques like recognizing sum of cubes are employed. The first polynomial has zeros at x = 0 and x = -5, while the second polynomial only has a zero at x = 0, as the cubic factor does not factor nicely.
Step-by-step explanation:
To factor the given polynomial functions and find all zeros, we first examine the polynomials one by one. For f(x) = x⁴ + 125x, we can factor out a common term of x, which gives us f(x) = x(x³ + 125). Recognizing that x³ + 125 is a sum of cubes, we can further factor it as x(x + 5)(x² - 5x + 25). To find the zeros we set each factor equal to zero: x = 0, x + 5 = 0, x² - 5x + 25 = 0. The quadratic factor does not have real solutions. Thus, the zeros are x = 0 and x = -5.
The second function f(x) = x⁴ + 5x³ + 4x² + 20x also has a common factor of x. So we factor it as f(x) = x(x³ + 5x² + 4x + 20). The cubic factor can be tried for synthetic division or using the rational root theorem, but it does not factor nicely over the integers, and we can't find any rational zeros. Hence, the only zero we can find easily for this function is x = 0.