Factor the following polynomial function completely. Use the Conjugate Roots Theorem, if applicable.
\[
f(x)=x^{6}-13 x^{4}-52 x^{2}+64
\]
Answer
\[
f(x)=
\]

Step 1 ：The given polynomial is a sextic (degree 6) polynomial. It is also a quadratic in \(x^2\). We can use the quadratic formula to find the roots of the polynomial, and then use the Conjugate Roots Theorem to find the remaining roots. The Conjugate Roots Theorem states that if a polynomial has real coefficients and a complex number is a root, then its conjugate is also a root.

Step 2 ：The polynomial has been factored completely. The roots of the polynomial are \(x = -4, -1, 1, 4\) and \(x = \pm 2i\). The roots \(x = \pm 2i\) are complex conjugates of each other, which is consistent with the Conjugate Roots Theorem.

Step 3 ：The factored form of the polynomial is \(\boxed{f(x) = (x - 4)(x - 1)(x + 1)(x + 4)(x^2 + 4)}\)