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 - 6x^4 - 31x^2 + 36

f(x) =

Solution

Step 1: Let \( y = x^2 \). \[ f(x) = y^3 - 6y^2 - 31y + 36 \] Step 2: Factor the cubic polynomial \( y^3 - 6y^2 - 31y + 36 \). \[ y^3 - 6y^2 - 31y + 36 = (y - 9)(y + 3)(y - 4) \] Step 3: Substitute back \( y = x^2 \). \[ (x^2 - 9)(x^2 + 3)(x^2 - 4) \] Step 4: Factor each quadratic further if possible. \[ (x - 3)(x + 3)(x^2 + 3)(x - 2)(x + 2) \] The completely factored form of the polynomial is: \[ (x - 3)(x + 3)(x^2 + 3)(x - 2)(x + 2) \]