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Factor \(\displaystyle - 12 x^{3} - 54 x^{2} - 6 x - 27\).

Factoring out the GCF \(\displaystyle -3\) from each term gives \(\displaystyle -3(4 x^{3} + 18 x^{2} + 2 x + 9)\). Grouping the first two terms and factoring out their GCF, \(\displaystyle 2 x^{2}\), gives \(\displaystyle 2 x^{2}(2 x + 9)\). Grouping the last two terms and factoring out their GCF, \(\displaystyle 1\), gives \(\displaystyle 1(2 x + 9)\). The polynomial now has a common binomial factor of \(\displaystyle 2 x + 9\). This gives \(\displaystyle -3[2 x^{2} \left(2 x + 9\right) +1 \cdot \left(2 x + 9\right)] = -3\left(2 x + 9\right) \left(2 x^{2} + 1\right)\).

Download \(\LaTeX\) 

\begin{question}Factor $- 12 x^{3} - 54 x^{2} - 6 x - 27$. 
    \soln{9cm}{Factoring out the GCF $-3$ from each term gives $-3(4 x^{3} + 18 x^{2} + 2 x + 9)$. Grouping the first two terms and factoring out their GCF, $2 x^{2}$, gives $2 x^{2}(2 x + 9)$. Grouping the last two terms and factoring out their GCF, $1$, gives $1(2 x + 9)$. The polynomial now has a common binomial factor of $2 x + 9$. This gives $-3[2 x^{2} \left(2 x + 9\right) +1 \cdot \left(2 x + 9\right)] = -3\left(2 x + 9\right) \left(2 x^{2} + 1\right)$. }

\end{question}

Download Question and Solution Environment\(\LaTeX\)
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HTML for Canvas 
<p> <p>Factor  <img class="equation_image" title=" \displaystyle - 12 x^{3} - 54 x^{2} - 6 x - 27 " src="/equation_images/%20%5Cdisplaystyle%20-%2012%20x%5E%7B3%7D%20-%2054%20x%5E%7B2%7D%20-%206%20x%20-%2027%20" alt="LaTeX:  \displaystyle - 12 x^{3} - 54 x^{2} - 6 x - 27 " data-equation-content=" \displaystyle - 12 x^{3} - 54 x^{2} - 6 x - 27 " /> . </p> </p>
HTML for Canvas
<p> <p>Factoring out the GCF  <img class="equation_image" title=" \displaystyle -3 " src="/equation_images/%20%5Cdisplaystyle%20-3%20" alt="LaTeX:  \displaystyle -3 " data-equation-content=" \displaystyle -3 " />  from each term gives  <img class="equation_image" title=" \displaystyle -3(4 x^{3} + 18 x^{2} + 2 x + 9) " src="/equation_images/%20%5Cdisplaystyle%20-3%284%20x%5E%7B3%7D%20%2B%2018%20x%5E%7B2%7D%20%2B%202%20x%20%2B%209%29%20" alt="LaTeX:  \displaystyle -3(4 x^{3} + 18 x^{2} + 2 x + 9) " data-equation-content=" \displaystyle -3(4 x^{3} + 18 x^{2} + 2 x + 9) " /> . Grouping the first two terms and factoring out their GCF,  <img class="equation_image" title=" \displaystyle 2 x^{2} " src="/equation_images/%20%5Cdisplaystyle%202%20x%5E%7B2%7D%20" alt="LaTeX:  \displaystyle 2 x^{2} " data-equation-content=" \displaystyle 2 x^{2} " /> , gives  <img class="equation_image" title=" \displaystyle 2 x^{2}(2 x + 9) " src="/equation_images/%20%5Cdisplaystyle%202%20x%5E%7B2%7D%282%20x%20%2B%209%29%20" alt="LaTeX:  \displaystyle 2 x^{2}(2 x + 9) " data-equation-content=" \displaystyle 2 x^{2}(2 x + 9) " /> . Grouping the last two terms and factoring out their GCF,  <img class="equation_image" title=" \displaystyle 1 " src="/equation_images/%20%5Cdisplaystyle%201%20" alt="LaTeX:  \displaystyle 1 " data-equation-content=" \displaystyle 1 " /> , gives  <img class="equation_image" title=" \displaystyle 1(2 x + 9) " src="/equation_images/%20%5Cdisplaystyle%201%282%20x%20%2B%209%29%20" alt="LaTeX:  \displaystyle 1(2 x + 9) " data-equation-content=" \displaystyle 1(2 x + 9) " /> . The polynomial now has a common binomial factor of  <img class="equation_image" title=" \displaystyle 2 x + 9 " src="/equation_images/%20%5Cdisplaystyle%202%20x%20%2B%209%20" alt="LaTeX:  \displaystyle 2 x + 9 " data-equation-content=" \displaystyle 2 x + 9 " /> . This gives  <img class="equation_image" title=" \displaystyle -3[2 x^{2} \left(2 x + 9\right) +1 \cdot \left(2 x + 9\right)] = -3\left(2 x + 9\right) \left(2 x^{2} + 1\right) " src="/equation_images/%20%5Cdisplaystyle%20-3%5B2%20x%5E%7B2%7D%20%5Cleft%282%20x%20%2B%209%5Cright%29%20%2B1%20%5Ccdot%20%5Cleft%282%20x%20%2B%209%5Cright%29%5D%20%3D%20-3%5Cleft%282%20x%20%2B%209%5Cright%29%20%5Cleft%282%20x%5E%7B2%7D%20%2B%201%5Cright%29%20" alt="LaTeX:  \displaystyle -3[2 x^{2} \left(2 x + 9\right) +1 \cdot \left(2 x + 9\right)] = -3\left(2 x + 9\right) \left(2 x^{2} + 1\right) " data-equation-content=" \displaystyle -3[2 x^{2} \left(2 x + 9\right) +1 \cdot \left(2 x + 9\right)] = -3\left(2 x + 9\right) \left(2 x^{2} + 1\right) " /> . </p> </p>