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Factor \(\displaystyle 42 x^{3} + 54 x^{2} - 63 x - 81\).

Factoring out the GCF \(\displaystyle 3\) from each term gives \(\displaystyle 3(14 x^{3} + 18 x^{2} - 21 x - 27)\). Grouping the first two terms and factoring out their GCF, \(\displaystyle 2 x^{2}\), gives \(\displaystyle 2 x^{2}(7 x + 9)\). Grouping the last two terms and factoring out their GCF, \(\displaystyle -3\), gives \(\displaystyle -3(7 x + 9)\). The polynomial now has a common binomial factor of \(\displaystyle 7 x + 9\). This gives \(\displaystyle 3[2 x^{2} \left(7 x + 9\right) -3 \cdot \left(7 x + 9\right)] = 3\left(7 x + 9\right) \left(2 x^{2} - 3\right)\).

Download \(\LaTeX\) 

\begin{question}Factor $42 x^{3} + 54 x^{2} - 63 x - 81$. 
    \soln{9cm}{Factoring out the GCF $3$ from each term gives $3(14 x^{3} + 18 x^{2} - 21 x - 27)$. Grouping the first two terms and factoring out their GCF, $2 x^{2}$, gives $2 x^{2}(7 x + 9)$. Grouping the last two terms and factoring out their GCF, $-3$, gives $-3(7 x + 9)$. The polynomial now has a common binomial factor of $7 x + 9$. This gives $3[2 x^{2} \left(7 x + 9\right) -3 \cdot \left(7 x + 9\right)] = 3\left(7 x + 9\right) \left(2 x^{2} - 3\right)$. }

\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} + 54 x^{2} - 63 x - 81 " src="/equation_images/%20%5Cdisplaystyle%2042%20x%5E%7B3%7D%20%2B%2054%20x%5E%7B2%7D%20-%2063%20x%20-%2081%20" alt="LaTeX:  \displaystyle 42 x^{3} + 54 x^{2} - 63 x - 81 " data-equation-content=" \displaystyle 42 x^{3} + 54 x^{2} - 63 x - 81 " /> . </p> </p>
HTML for Canvas
<p> <p>Factoring out the GCF  <img class="equation_image" title=" \displaystyle 3 " src="/equation_images/%20%5Cdisplaystyle%203%20" alt="LaTeX:  \displaystyle 3 " data-equation-content=" \displaystyle 3 " />  from each term gives  <img class="equation_image" title=" \displaystyle 3(14 x^{3} + 18 x^{2} - 21 x - 27) " src="/equation_images/%20%5Cdisplaystyle%203%2814%20x%5E%7B3%7D%20%2B%2018%20x%5E%7B2%7D%20-%2021%20x%20-%2027%29%20" alt="LaTeX:  \displaystyle 3(14 x^{3} + 18 x^{2} - 21 x - 27) " data-equation-content=" \displaystyle 3(14 x^{3} + 18 x^{2} - 21 x - 27) " /> . 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 + 9) " src="/equation_images/%20%5Cdisplaystyle%202%20x%5E%7B2%7D%287%20x%20%2B%209%29%20" alt="LaTeX:  \displaystyle 2 x^{2}(7 x + 9) " data-equation-content=" \displaystyle 2 x^{2}(7 x + 9) " /> . Grouping the last two terms and factoring out their 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 " /> , gives  <img class="equation_image" title=" \displaystyle -3(7 x + 9) " src="/equation_images/%20%5Cdisplaystyle%20-3%287%20x%20%2B%209%29%20" alt="LaTeX:  \displaystyle -3(7 x + 9) " data-equation-content=" \displaystyle -3(7 x + 9) " /> . The polynomial now has a common binomial factor of  <img class="equation_image" title=" \displaystyle 7 x + 9 " src="/equation_images/%20%5Cdisplaystyle%207%20x%20%2B%209%20" alt="LaTeX:  \displaystyle 7 x + 9 " data-equation-content=" \displaystyle 7 x + 9 " /> . This gives  <img class="equation_image" title=" \displaystyle 3[2 x^{2} \left(7 x + 9\right) -3 \cdot \left(7 x + 9\right)] = 3\left(7 x + 9\right) \left(2 x^{2} - 3\right) " src="/equation_images/%20%5Cdisplaystyle%203%5B2%20x%5E%7B2%7D%20%5Cleft%287%20x%20%2B%209%5Cright%29%20-3%20%5Ccdot%20%5Cleft%287%20x%20%2B%209%5Cright%29%5D%20%3D%203%5Cleft%287%20x%20%2B%209%5Cright%29%20%5Cleft%282%20x%5E%7B2%7D%20-%203%5Cright%29%20" alt="LaTeX:  \displaystyle 3[2 x^{2} \left(7 x + 9\right) -3 \cdot \left(7 x + 9\right)] = 3\left(7 x + 9\right) \left(2 x^{2} - 3\right) " data-equation-content=" \displaystyle 3[2 x^{2} \left(7 x + 9\right) -3 \cdot \left(7 x + 9\right)] = 3\left(7 x + 9\right) \left(2 x^{2} - 3\right) " /> . </p> </p>