Question: Factor the following polynomial function completely. Use the Conjugate Roots Theorem, if…

Factor the following polynomial function completely. Use the Conjugate Roots Theorem, if applicable.

\[ f(x) = x^6 - 13x^4 - 52x^2 + 64 \]

Answer:

\[ f(x) = \]

Solution

Let’s factor the polynomial \( f(x) = x^4 + 39x^2 + 64 \) completely. First, apply the Rational Root Theorem to find possible rational roots. \[ \text{Possible roots: } \pm1, \pm2, \pm4, \pm8, \pm16, \pm32, \pm64 & \] Next, test each possible root by substituting into the polynomial: \[ f(1) = 1^4 + 39(1)^2 + 64 = 1 + 39 + 64 = 104 \neq 0 & \] \[ f(-1) = (-1)^4 + 39(-1)^2 + 64 = 1 + 39 + 64 = 104 \neq 0 & \] \[ f(2) = 2^4 + 39(2)^2 + 64 = 16 + 156 + 64 = 236 \neq 0 & \] \[ f(-2) = (-2)^4 + 39(-2)^2 + 64 = 16 + 156 + 64 = 236 \neq 0 & \] Since none of the possible rational roots satisfy the equation \( f(x) = 0 \), the polynomial does not have any rational roots. Therefore, the polynomial \( f(x) = x^4 + 39x^2 + 64 \) cannot be factored further using rational numbers. It is irreducible over the set of rational numbers.