Question: Given that \(2 - 2i\) is a zero, factor the following polynomial function completely. Use the…

Given that \(2 - 2i\) is a zero, factor the following polynomial function completely. Use the Conjugate Roots Theorem, if applicable.

\[ f(x) = x^4 + 3x^3 - 8x^2 + 8x + 96 \]

Solution

To factor the polynomial \( f(x) = x^4 + 3x^3 - 8x^2 + 8x + 96 \) completely, given that \( 2 - 2i \) is a zero, follow these steps: Step 1: Apply the Conjugate Roots Theorem Since \( 2 - 2i \) is a zero, its conjugate \( 2 + 2i \) is also a zero. Step 2: Form the Quadratic Factor from the Complex Zeros \[ (x - (2 - 2i))(x - (2 + 2i)) & \] \[ = (x - 2 + 2i)(x - 2 - 2i) & \] \[ = (x - 2)^2 - (2i)^2 & \] \[ = x^2 - 4x + 4 + 4 & \] \[ = x^2 - 4x + 8 & \] Step 3: Perform Polynomial Division Divide \( f(x) \) by \( x^2 - 4x + 8 \) to find the remaining quadratic factor. \[ f(x) = (x^2 - 4x + 8)(x^2 + ax + b) & \] Expanding the right side: \[ x^4 + ax^3 + bx^2 - 4x^3 - 4ax^2 - 4bx + 8x^2 + 8ax + 8b & \] Combine like terms: \[ x^4 + (a - 4)x^3 + (b - 4a + 8)x^2 + (-4b + 8a)x + 8b & \] Set the coefficients equal to those of \( f(x) \): \[ a - 4 = 3 \\ a = 7 & \] \[ b - 4a + 8 = -8 \\ b - 28 + 8 = -8 \\ b - 20 = -8 \\ b = 12 & \] Step 4: Factor the Remaining Quadratic \[ x^2 + 7x + 12 & \] \[ = (x + 3)(x + 4) & \] Final Factorization \[ f(x) = (x^2 - 4x + 8)(x + 3)(x + 4) & \] Alternatively, expressed with all linear factors (including complex zeros): \[ f(x) = (x - (2 - 2i))(x - (2 + 2i))(x + 3)(x + 4) & \]