Final answer:
To determine all complex zeros, set the function equal to zero and solve for x. The function g(x) has a total of five complex zeros.
Step-by-step explanation:
To determine all complex zeros of the function g(x) = x^5 + x^4 - 5x^3 + x^2 - 6x, we set the function equal to zero and solve for x. We can do this by factoring out an x from the equation:
x(x^4 + x^3 - 5x^2 + x - 6) = 0
From the factored form, we can see that there is a complex zero at x = 0.
Next, we set the expression inside the parentheses equal to zero and solve for the remaining complex zeros. This can be done using various methods such as synthetic division or factoring by grouping.
After solving for the complex zeros, we find that the function g(x) has a total of five complex zeros.