Question: Consider the following polynomial. \[ G(x) = x^3 - 5x^2 - 11x + 55 \] Step 2 of 2: Use…

Consider the following polynomial.

\[ G(x) = x^3 - 5x^2 - 11x + 55 \]

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

Find the zeros of the polynomial \( G(x) = x^3 - 5x^2 - 11x + 55 \). Start by applying the Rational Root Theorem. \[ \text{Possible rational roots are } \pm1, \pm5, \pm11, \pm55. \]& Test \( x = 5 \): \[ G(5) = 5^3 - 5(5)^2 - 11(5) + 55 = 125 - 125 - 55 + 55 = 0 \]& Since \( G(5) = 0 \), \( x = 5 \) is a root. Next, perform polynomial division to factor out \( (x - 5) \) from \( G(x) \): \[ G(x) = (x - 5)(x^2 - 11) \]& Now, solve the quadratic equation \( x^2 - 11 = 0 \): \[ x^2 - 11 = 0 \implies x = \pm \sqrt{11} \]& Therefore, the zeros of \( G(x) \) are: \[ 5, \sqrt{11}, -\sqrt{11} \]