Final answer:
To find the zeros of the given function f(x), we use polynomial long division or synthetic division to find the other factor. The zeros of the function are x = -4 and x = 2.
Step-by-step explanation:
To find the zeros of the function f(x), we need to solve the equation f(x) = 0. Given that f(x) = x^3 + 5x^2 + 7x - 13x - 1 is a factor of f(x), we can use polynomial long division or synthetic division to find the other factor.
Performing synthetic division, we divide f(x) by x - 1:
1│ 1 5 7 -13 -1
────── -4 1 8 -5
1 1 8 -5 | 3
The quotient is x^2 + x + 8 - 5/(x - 1) = x^2 + x + 8 - 5/(x - 1). Setting this quotient equal to zero and solving for x, we get:
x^2 + x + 8 - 5/(x - 1) = 0.
By factoring the quadratic equation, we find the zeros of the function to be:
x^2 + x + 8 - 5/(x - 1) = 0
(x + 4)(x - 2) = 0
x = -4 or x = 2