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Factor \(\displaystyle 27 x^{3} + 21 x^{2} - 36 x - 28\).

Grouping the first two terms and factoring out their GCF, \(\displaystyle 3 x^{2}\), gives \(\displaystyle 3 x^{2}(9 x + 7)\). Grouping the last two terms and factoring out their GCF, \(\displaystyle -4\), gives \(\displaystyle -4(9 x + 7)\). The polynomial now has a common binomial factor of \(\displaystyle 9 x + 7\). This gives \(\displaystyle 3 x^{2} \left(9 x + 7\right) -4 \cdot \left(9 x + 7\right) = \left(9 x + 7\right) \left(3 x^{2} - 4\right)\).

Download \(\LaTeX\) 

\begin{question}Factor $27 x^{3} + 21 x^{2} - 36 x - 28$. 
    \soln{9cm}{Grouping the first two terms and factoring out their GCF, $3 x^{2}$, gives $3 x^{2}(9 x + 7)$. Grouping the last two terms and factoring out their GCF, $-4$, gives $-4(9 x + 7)$. The polynomial now has a common binomial factor of $9 x + 7$. This gives $3 x^{2} \left(9 x + 7\right) -4 \cdot \left(9 x + 7\right) = \left(9 x + 7\right) \left(3 x^{2} - 4\right)$. }

\end{question}

Download Question and Solution Environment\(\LaTeX\)
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HTML for Canvas 
<p> <p>Factor  <img class="equation_image" title=" \displaystyle 27 x^{3} + 21 x^{2} - 36 x - 28 " src="/equation_images/%20%5Cdisplaystyle%2027%20x%5E%7B3%7D%20%2B%2021%20x%5E%7B2%7D%20-%2036%20x%20-%2028%20" alt="LaTeX:  \displaystyle 27 x^{3} + 21 x^{2} - 36 x - 28 " data-equation-content=" \displaystyle 27 x^{3} + 21 x^{2} - 36 x - 28 " /> . </p> </p>
HTML for Canvas
<p> <p>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 + 7) " src="/equation_images/%20%5Cdisplaystyle%203%20x%5E%7B2%7D%289%20x%20%2B%207%29%20" alt="LaTeX:  \displaystyle 3 x^{2}(9 x + 7) " data-equation-content=" \displaystyle 3 x^{2}(9 x + 7) " /> . Grouping the last two terms and factoring out their GCF,  <img class="equation_image" title=" \displaystyle -4 " src="/equation_images/%20%5Cdisplaystyle%20-4%20" alt="LaTeX:  \displaystyle -4 " data-equation-content=" \displaystyle -4 " /> , gives  <img class="equation_image" title=" \displaystyle -4(9 x + 7) " src="/equation_images/%20%5Cdisplaystyle%20-4%289%20x%20%2B%207%29%20" alt="LaTeX:  \displaystyle -4(9 x + 7) " data-equation-content=" \displaystyle -4(9 x + 7) " /> . The polynomial now has a common binomial factor of  <img class="equation_image" title=" \displaystyle 9 x + 7 " src="/equation_images/%20%5Cdisplaystyle%209%20x%20%2B%207%20" alt="LaTeX:  \displaystyle 9 x + 7 " data-equation-content=" \displaystyle 9 x + 7 " /> . This gives  <img class="equation_image" title=" \displaystyle 3 x^{2} \left(9 x + 7\right) -4 \cdot \left(9 x + 7\right) = \left(9 x + 7\right) \left(3 x^{2} - 4\right) " src="/equation_images/%20%5Cdisplaystyle%203%20x%5E%7B2%7D%20%5Cleft%289%20x%20%2B%207%5Cright%29%20-4%20%5Ccdot%20%5Cleft%289%20x%20%2B%207%5Cright%29%20%3D%20%5Cleft%289%20x%20%2B%207%5Cright%29%20%5Cleft%283%20x%5E%7B2%7D%20-%204%5Cright%29%20" alt="LaTeX:  \displaystyle 3 x^{2} \left(9 x + 7\right) -4 \cdot \left(9 x + 7\right) = \left(9 x + 7\right) \left(3 x^{2} - 4\right) " data-equation-content=" \displaystyle 3 x^{2} \left(9 x + 7\right) -4 \cdot \left(9 x + 7\right) = \left(9 x + 7\right) \left(3 x^{2} - 4\right) " /> . </p> </p>