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Factor \(\displaystyle 54 x^{3} + 12 x^{2} + 36 x + 8\).

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

Download \(\LaTeX\) 

\begin{question}Factor $54 x^{3} + 12 x^{2} + 36 x + 8$. 
    \soln{9cm}{Factoring out the GCF $2$ from each term gives $2(27 x^{3} + 6 x^{2} + 18 x + 4)$. Grouping the first two terms and factoring out their GCF, $3 x^{2}$, gives $3 x^{2}(9 x + 2)$. Grouping the last two terms and factoring out their GCF, $2$, gives $2(9 x + 2)$. The polynomial now has a common binomial factor of $9 x + 2$. This gives $2[3 x^{2} \left(9 x + 2\right) +2 \cdot \left(9 x + 2\right)] = 2\left(9 x + 2\right) \left(3 x^{2} + 2\right)$. }

\end{question}

Download Question and Solution Environment\(\LaTeX\)
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HTML for Canvas 
<p> <p>Factor  <img class="equation_image" title=" \displaystyle 54 x^{3} + 12 x^{2} + 36 x + 8 " src="/equation_images/%20%5Cdisplaystyle%2054%20x%5E%7B3%7D%20%2B%2012%20x%5E%7B2%7D%20%2B%2036%20x%20%2B%208%20" alt="LaTeX:  \displaystyle 54 x^{3} + 12 x^{2} + 36 x + 8 " data-equation-content=" \displaystyle 54 x^{3} + 12 x^{2} + 36 x + 8 " /> . </p> </p>
HTML for Canvas
<p> <p>Factoring out the GCF  <img class="equation_image" title=" \displaystyle 2 " src="/equation_images/%20%5Cdisplaystyle%202%20" alt="LaTeX:  \displaystyle 2 " data-equation-content=" \displaystyle 2 " />  from each term gives  <img class="equation_image" title=" \displaystyle 2(27 x^{3} + 6 x^{2} + 18 x + 4) " src="/equation_images/%20%5Cdisplaystyle%202%2827%20x%5E%7B3%7D%20%2B%206%20x%5E%7B2%7D%20%2B%2018%20x%20%2B%204%29%20" alt="LaTeX:  \displaystyle 2(27 x^{3} + 6 x^{2} + 18 x + 4) " data-equation-content=" \displaystyle 2(27 x^{3} + 6 x^{2} + 18 x + 4) " /> . Grouping the first two terms and factoring out their GCF,  <img class="equation_image" title=" \displaystyle 3 x^{2} " src="/equation_images/%20%5Cdisplaystyle%203%20x%5E%7B2%7D%20" alt="LaTeX:  \displaystyle 3 x^{2} " data-equation-content=" \displaystyle 3 x^{2} " /> , gives  <img class="equation_image" title=" \displaystyle 3 x^{2}(9 x + 2) " src="/equation_images/%20%5Cdisplaystyle%203%20x%5E%7B2%7D%289%20x%20%2B%202%29%20" alt="LaTeX:  \displaystyle 3 x^{2}(9 x + 2) " data-equation-content=" \displaystyle 3 x^{2}(9 x + 2) " /> . Grouping the last two terms and factoring out their GCF,  <img class="equation_image" title=" \displaystyle 2 " src="/equation_images/%20%5Cdisplaystyle%202%20" alt="LaTeX:  \displaystyle 2 " data-equation-content=" \displaystyle 2 " /> , gives  <img class="equation_image" title=" \displaystyle 2(9 x + 2) " src="/equation_images/%20%5Cdisplaystyle%202%289%20x%20%2B%202%29%20" alt="LaTeX:  \displaystyle 2(9 x + 2) " data-equation-content=" \displaystyle 2(9 x + 2) " /> . The polynomial now has a common binomial factor of  <img class="equation_image" title=" \displaystyle 9 x + 2 " src="/equation_images/%20%5Cdisplaystyle%209%20x%20%2B%202%20" alt="LaTeX:  \displaystyle 9 x + 2 " data-equation-content=" \displaystyle 9 x + 2 " /> . This gives  <img class="equation_image" title=" \displaystyle 2[3 x^{2} \left(9 x + 2\right) +2 \cdot \left(9 x + 2\right)] = 2\left(9 x + 2\right) \left(3 x^{2} + 2\right) " src="/equation_images/%20%5Cdisplaystyle%202%5B3%20x%5E%7B2%7D%20%5Cleft%289%20x%20%2B%202%5Cright%29%20%2B2%20%5Ccdot%20%5Cleft%289%20x%20%2B%202%5Cright%29%5D%20%3D%202%5Cleft%289%20x%20%2B%202%5Cright%29%20%5Cleft%283%20x%5E%7B2%7D%20%2B%202%5Cright%29%20" alt="LaTeX:  \displaystyle 2[3 x^{2} \left(9 x + 2\right) +2 \cdot \left(9 x + 2\right)] = 2\left(9 x + 2\right) \left(3 x^{2} + 2\right) " data-equation-content=" \displaystyle 2[3 x^{2} \left(9 x + 2\right) +2 \cdot \left(9 x + 2\right)] = 2\left(9 x + 2\right) \left(3 x^{2} + 2\right) " /> . </p> </p>