Final answer:
To find the other zeros of the given polynomial function, use the Rational Root Theorem to find possible rational zeros. Then, divide the function by the known zero to obtain the remaining quadratic factor. Finally, solve the quadratic equation to find the other zeros.
Step-by-step explanation:
To find the other zeros of the given polynomial function f(x) = x^3 + 3x^2 - 34x + 48, we can use the Rational Root Theorem. This theorem states that if a rational number p/q is a zero of a polynomial function, then p must be a factor of the constant term and q must be a factor of the leading coefficient. In this case, the constant term is 48 and the leading coefficient is 1. So, the possible rational zeros are ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±16, ±24, ±48.
By substituting these values into the function, we can see that the rational zero of f(x) is -2. Now, we can use synthetic division or long division to divide f(x) by (x - (-2)) or (x + 2) to find the remaining quadratic factor. The result is x^2 + 5x - 24. To find the other zeros, we can solve the equation x^2 + 5x - 24 = 0 using factoring, completing the square, or using the quadratic formula. The solutions are x = -8 and x = 3.