Final answer:
To find all zeros of f(x), confirm that -5 is a zero, then divide f(x) by (x + 5) to get a quadratic equation. Solve this quadratic using the quadratic formula to find the remaining zeros.
Step-by-step explanation:
To find all of the zeros of the function algebraically, given that f(x) = x^3 + 2x^2 - 13x + 10 and f(-5) = 0, first, we confirm that -5 is a zero by substitution. Since f(-5) = 0, we know that (x + 5) is a factor of the function. We can use polynomial long division or synthetic division to divide f(x) by (x + 5) to get the other factors. The result will be a quadratic equation, which we can then solve using the quadratic formula. The quadratic formula is x = (-b ± √(b^2 - 4ac))/(2a), where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c = 0. After finding the two other zeroes from this quadratic equation, we will have all the zeros of f(x).