Find all complex zeros of the polynomial function Give exact values. List multiple zeros as necessary. f(x)=x^5-9x^4+30x^3-54x^2+81x-81 All complex zeros are square (Type an exact answer using radicals and i as needed. Use a comma to separate answers as needed.)

To find the complex zeros of the polynomial function \( f(x) = x^5 - 9x^4 + 30x^3 - 54x^2 + 81x - 81 \), we can use the Rational Root Theorem and synthetic division to identify possible rational roots and then factor the polynomial.First, let's use the Rational Root Theorem to identify possible rational roots. The Rational Root Theorem states that any rational root of the polynomial \( f(x) \) is a factor of the constant term (-81) divided by a factor of the leading coefficient (1). Therefore, the possible rational roots are: We will test these possible roots by substituting them into the polynomial to see if they yield zero.Let's start with : Since \( f(1) = 0 \), is a root of the polynomial. We can now perform synthetic division to factor out \( (x - 1) \) from \( f(x) \).Using synthetic division with : The quotient is . So, we have: Next, we need to find the roots of the quartic polynomial . We can use the quadratic formula to solve this quartic polynomial. Let , then the quartic polynomial becomes: Using the quadratic formula : So, . Since , we have: Taking the square roots of these: Thus, the complex zeros of the polynomial \( f(x) = x^5 - 9x^4 + 30x^3 - 54x^2 + 81x - 81 \) are: All complex zeros are: