Question

factor the polynomial completely. -3w^3 - 6w^2 + 27w + 54 select the correct choice below and fill in any answer boxes within your choice. a. -3w^3 - 6w^2 + 27w + 54 = b. the polynomial is prime.

Explanation:

Step1: Factor out the greatest - common factor

First, find the GCF of the coefficients \(-3,-6,27,54\). The GCF is \(-3\).
\[

$$\begin{align*} -3w^{3}-6w^{2}+27w + 54&=-3(w^{3}+2w^{2}-9w - 18) \end{align*}$$

\]

Step2: Group the terms

Group the terms inside the parentheses: \((w^{3}+2w^{2})+(-9w - 18)\)

Step3: Factor each group

Factor out \(w^{2}\) from the first group and \(-9\) from the second group:
\[

$$\begin{align*} (w^{3}+2w^{2})+(-9w - 18)&=w^{2}(w + 2)-9(w + 2) \end{align*}$$

\]

Step4: Factor out the common binomial factor

Factor out \((w + 2)\) from the above expression:
\[

$$\begin{align*} w^{2}(w + 2)-9(w + 2)&=(w + 2)(w^{2}-9) \end{align*}$$

\]

Step5: Factor the difference of squares

Since \(w^{2}-9=(w + 3)(w - 3)\) (because \(a^{2}-b^{2}=(a + b)(a - b)\) with \(a = w\) and \(b = 3\)), the completely factored form is \(-3(w + 2)(w+3)(w - 3)\)

Answer:

A. \(-3(w + 2)(w + 3)(w - 3)\)