Question: Consider the following polynomial function. \[ f(x) = x^4 + x^3 - 11x^2 - 5x + 30 \] Step 1 of…

Consider the following polynomial function.

\[ f(x) = x^4 + x^3 - 11x^2 - 5x + 30 \]

Step 1 of 4: Factor the polynomial completely.

Answer

\[ f(x) = \]

Solution

To factor the polynomial completely, follow these steps: Step 1: Find rational roots using the Rational Root Theorem. The possible rational roots are the factors of the constant term divided by the factors of the leading coefficient. \[ \text{Possible roots} = \pm1, \pm2, \pm3, \pm5, \pm6, \pm10, \pm15, \pm30 & \] Step 2: Test the possible roots to find actual roots. Test \( x = 2 \): \[ f(2) = (2)^4 + (2)^3 - 11(2)^2 - 5(2) + 30 = 16 + 8 - 44 - 10 + 30 = 0 & \] So, \( x = 2 \) is a root. Step 3: Perform polynomial division by \( (x - 2) \). Divide \( f(x) \) by \( (x - 2) \): \[ f(x) = (x - 2)(x^3 + 3x^2 - 5x - 15) & \] Step 4: Factor the cubic polynomial \( x^3 + 3x^2 - 5x - 15 \). Find a root of \( x^3 + 3x^2 - 5x - 15 \): Test \( x = 3 \): \[ 3^3 + 3(3)^2 - 5(3) - 15 = 27 + 27 - 15 - 15 = 24 \neq 0 & \] Test \( x = -3 \): \[ (-3)^3 + 3(-3)^2 - 5(-3) - 15 = -27 + 27 + 15 - 15 = 0 & \] So, \( x = -3 \) is a root. Step 5: Perform polynomial division by \( (x + 3) \). Divide \( x^3 + 3x^2 - 5x - 15 \) by \( (x + 3) \): \[ x^3 + 3x^2 - 5x - 15 = (x + 3)(x^2 - 5) & \] Step 6: Factor \( x^2 - 5 \). \[ x^2 - 5 = (x - \sqrt{5})(x + \sqrt{5}) & \] Final Factored Form: \[ f(x) = (x - 2)(x + 3)(x - \sqrt{5})(x + \sqrt{5}) \]