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Factor \(\displaystyle - 42 x^{3} - 36 x^{2} - 63 x - 54\).

Factoring out the GCF \(\displaystyle -3\) from each term gives \(\displaystyle -3(14 x^{3} + 12 x^{2} + 21 x + 18)\). Grouping the first two terms and factoring out their GCF, \(\displaystyle 2 x^{2}\), gives \(\displaystyle 2 x^{2}(7 x + 6)\). Grouping the last two terms and factoring out their GCF, \(\displaystyle 3\), gives \(\displaystyle 3(7 x + 6)\). The polynomial now has a common binomial factor of \(\displaystyle 7 x + 6\). This gives \(\displaystyle -3[2 x^{2} \left(7 x + 6\right) +3 \cdot \left(7 x + 6\right)] = -3\left(7 x + 6\right) \left(2 x^{2} + 3\right)\).

Download \(\LaTeX\) 

\begin{question}Factor $- 42 x^{3} - 36 x^{2} - 63 x - 54$. 
    \soln{9cm}{Factoring out the GCF $-3$ from each term gives $-3(14 x^{3} + 12 x^{2} + 21 x + 18)$. Grouping the first two terms and factoring out their GCF, $2 x^{2}$, gives $2 x^{2}(7 x + 6)$. Grouping the last two terms and factoring out their GCF, $3$, gives $3(7 x + 6)$. The polynomial now has a common binomial factor of $7 x + 6$. This gives $-3[2 x^{2} \left(7 x + 6\right) +3 \cdot \left(7 x + 6\right)] = -3\left(7 x + 6\right) \left(2 x^{2} + 3\right)$. }

\end{question}

Download Question and Solution Environment\(\LaTeX\)
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HTML for Canvas 
<p> <p>Factor  <img class="equation_image" title=" \displaystyle - 42 x^{3} - 36 x^{2} - 63 x - 54 " src="/equation_images/%20%5Cdisplaystyle%20-%2042%20x%5E%7B3%7D%20-%2036%20x%5E%7B2%7D%20-%2063%20x%20-%2054%20" alt="LaTeX:  \displaystyle - 42 x^{3} - 36 x^{2} - 63 x - 54 " data-equation-content=" \displaystyle - 42 x^{3} - 36 x^{2} - 63 x - 54 " /> . </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(14 x^{3} + 12 x^{2} + 21 x + 18) " src="/equation_images/%20%5Cdisplaystyle%20-3%2814%20x%5E%7B3%7D%20%2B%2012%20x%5E%7B2%7D%20%2B%2021%20x%20%2B%2018%29%20" alt="LaTeX:  \displaystyle -3(14 x^{3} + 12 x^{2} + 21 x + 18) " data-equation-content=" \displaystyle -3(14 x^{3} + 12 x^{2} + 21 x + 18) " /> . 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}(7 x + 6) " src="/equation_images/%20%5Cdisplaystyle%202%20x%5E%7B2%7D%287%20x%20%2B%206%29%20" alt="LaTeX:  \displaystyle 2 x^{2}(7 x + 6) " data-equation-content=" \displaystyle 2 x^{2}(7 x + 6) " /> . Grouping the last two terms and factoring out their GCF,  <img class="equation_image" title=" \displaystyle 3 " src="/equation_images/%20%5Cdisplaystyle%203%20" alt="LaTeX:  \displaystyle 3 " data-equation-content=" \displaystyle 3 " /> , gives  <img class="equation_image" title=" \displaystyle 3(7 x + 6) " src="/equation_images/%20%5Cdisplaystyle%203%287%20x%20%2B%206%29%20" alt="LaTeX:  \displaystyle 3(7 x + 6) " data-equation-content=" \displaystyle 3(7 x + 6) " /> . The polynomial now has a common binomial factor of  <img class="equation_image" title=" \displaystyle 7 x + 6 " src="/equation_images/%20%5Cdisplaystyle%207%20x%20%2B%206%20" alt="LaTeX:  \displaystyle 7 x + 6 " data-equation-content=" \displaystyle 7 x + 6 " /> . This gives  <img class="equation_image" title=" \displaystyle -3[2 x^{2} \left(7 x + 6\right) +3 \cdot \left(7 x + 6\right)] = -3\left(7 x + 6\right) \left(2 x^{2} + 3\right) " src="/equation_images/%20%5Cdisplaystyle%20-3%5B2%20x%5E%7B2%7D%20%5Cleft%287%20x%20%2B%206%5Cright%29%20%2B3%20%5Ccdot%20%5Cleft%287%20x%20%2B%206%5Cright%29%5D%20%3D%20-3%5Cleft%287%20x%20%2B%206%5Cright%29%20%5Cleft%282%20x%5E%7B2%7D%20%2B%203%5Cright%29%20" alt="LaTeX:  \displaystyle -3[2 x^{2} \left(7 x + 6\right) +3 \cdot \left(7 x + 6\right)] = -3\left(7 x + 6\right) \left(2 x^{2} + 3\right) " data-equation-content=" \displaystyle -3[2 x^{2} \left(7 x + 6\right) +3 \cdot \left(7 x + 6\right)] = -3\left(7 x + 6\right) \left(2 x^{2} + 3\right) " /> . </p> </p>