Find all complex zeros of the given polynomial function, and write the polynomial in completely factored form.

[tex] f(x) = -3x^5 + 35x^4 - 129x^3 + 139x^2 - 16x - 26 [/tex]

Find the complex zeros of [tex] f(x) [/tex]. Repeat any zeros if their multiplicity is greater than 1.

[tex] x = \square [/tex]

(Simplify your answer. Use a comma to separate answers as needed. Express complex numbers in terms of [tex] i [/tex].)

To find the complex zeros of the polynomial function [tex]$ f(x) = -3x^5 + 35x^4 - 129x^3 + 139x^2 - 16x - 26 $[/tex] and write the polynomial in completely factored form, follow these steps:

1. Identify the Zeros:
Based on the results, the complex zeros of the polynomial are:
- [tex]$ x = -\frac{1}{3} $[/tex]
- [tex]$ x = 1 $[/tex]
- [tex]$ x = 5 - i $[/tex]
- [tex]$ x = 5 + i $[/tex]

2. Understand the Multiplicity:
Each of these zeros appears once, so they have a multiplicity of 1.

3. Write the Polynomial in Factor Form:
Knowing the zeros, we can express the polynomial in its completely factored form:

[tex]\[
f(x) = -3 \left( x + \frac{1}{3} \right) (x - 1) (x - (5 - i)) (x - (5 + i))
\][/tex]

4. Simplify the Factored Form:
To simplify, start by adjusting the factor [tex]$ x + \frac{1}{3} $[/tex] to avoid fractions:

[tex]\[
-3\left(x + \frac{1}{3}\right) = (-3)(x + \frac{1}{3}) = -3x - 1
\][/tex]

The final completely factored form is:

[tex]\[
f(x) = (-3x - 1)(x - 1)(x - (5 - i))(x - (5 + i))
\][/tex]

This gives us the polynomial expressed in factored form showcasing all its complex zeros.