Question: Consider the following polynomial. \[ G(x) = x^4 - 5x^3 + 2x^2 + 20x - 24 \] Step 2 of 2: Use…

Consider the following polynomial.

\[ G(x) = x^4 - 5x^3 + 2x^2 + 20x - 24 \]

Step 2 of 2: Use polynomial division and the quadratic formula, if necessary, to identify the actual zeros.

Answer

Separate multiple answers with commas.

Solution

Solution: First, use the Rational Root Theorem to identify possible rational zeros. \[ \text{Possible zeros: } \pm1, \pm2, \pm3, \pm4, \pm6, \pm12, \pm24 \] Next, test \( x = 2 \): \[ G(2) = 2^4 - 5(2)^3 + 2(2)^2 + 20(2) - 24 = 0 \] Since \( G(2) = 0 \), \( x = 2 \) is a zero. Perform polynomial division by \( (x - 2) \): \[ G(x) = (x - 2)(x^3 - 3x^2 - x + 12) \] Now, find zeros of \( x^3 - 3x^2 - x + 12 \). Test \( x = 3 \): \[ 3^3 - 3(3)^2 - 3 + 12 = 0 \] Since \( x = 3 \) is a zero, perform division by \( (x - 3) \): \[ x^3 - 3x^2 - x + 12 = (x - 3)(x^2 - 4) \] Finally, solve \( x^2 - 4 = 0 \): \[ \begin{cases} x - 2 = 0 & \\ x + 2 = 0 & \end{cases} \] Thus, the zeros of \( G(x) \) are: \[ 2, \, 3, \, -2 \]