Find the complex zeros of the polynomial function, and write f in factored form:

f(x)= x^3 - 8x^2 + 30x - 36

any real roots will be 1,2,3,4,6,9,12,24,36

Since the inner coefficients are small, I'd start with small values. A little synthetic division yields

(x-2)(x^2 - 6x + 18)

Now just use the quadratic formula to find the complex roots.

To find the complex zeros of the polynomial function f(x) = x^3 - 8x^2 + 30x - 36 and write it in factored form, we can use the Rational Root Theorem and synthetic division.

Step 1: List all possible rational roots
The Rational Root Theorem states that the rational roots of a polynomial are of the form p/q, where p is a factor of the constant term (-36) and q is a factor of the leading coefficient (1 in this case). Since the constant term is -36 and the leading coefficient is 1, the possible rational roots are ±1, ±2, ±3, ±4, ±6, ±9, ±12, ±18, and ±36.

Step 2: Use synthetic division to check for roots
We can use synthetic division to test the possible rational roots and find any that result in a remainder of 0.

Let's start by testing the root x = 1:
1 | 1 -8 30 -36
| 1 -7 23 -13
---------------
1 -7 23 -13

The remainder is -13, which is not zero. Therefore, x = 1 is not a root of the polynomial.

Next, let's test the root x = -1:
-1 | 1 -8 30 -36
| -1 9 -39 9
--------------------
1 -7 -9 -27

The remainder is -27, which is not zero. Therefore, x = -1 is not a root of the polynomial.

Repeat this process for all the possible rational roots to find the roots of the polynomial.

Step 3: Use the found roots to write f(x) in factored form
After testing all the possible rational roots, we find that the polynomial does not have any rational roots. Therefore, there are no complex zeros and the polynomial cannot be factored further.

So, the complex zeros of the polynomial function f(x) = x^3 - 8x^2 + 30x - 36 are not found and the polynomial is already in its factored form as f(x) = (x^3 - 8x^2 + 30x - 36).