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Factor \(\displaystyle 27 x^{3} - 63 x^{2} - 15 x + 35\).

Grouping the first two terms and factoring out their GCF, \(\displaystyle 9 x^{2}\), gives \(\displaystyle 9 x^{2}(3 x - 7)\). Grouping the last two terms and factoring out their GCF, \(\displaystyle -5\), gives \(\displaystyle -5(3 x - 7)\). The polynomial now has a common binomial factor of \(\displaystyle 3 x - 7\). This gives \(\displaystyle 9 x^{2} \left(3 x - 7\right) -5 \cdot \left(3 x - 7\right) = \left(3 x - 7\right) \left(9 x^{2} - 5\right)\).

Download \(\LaTeX\) 

\begin{question}Factor $27 x^{3} - 63 x^{2} - 15 x + 35$. 
    \soln{9cm}{Grouping the first two terms and factoring out their GCF, $9 x^{2}$, gives $9 x^{2}(3 x - 7)$. Grouping the last two terms and factoring out their GCF, $-5$, gives $-5(3 x - 7)$. The polynomial now has a common binomial factor of $3 x - 7$. This gives $9 x^{2} \left(3 x - 7\right) -5 \cdot \left(3 x - 7\right) = \left(3 x - 7\right) \left(9 x^{2} - 5\right)$. }

\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} - 63 x^{2} - 15 x + 35 " src="/equation_images/%20%5Cdisplaystyle%2027%20x%5E%7B3%7D%20-%2063%20x%5E%7B2%7D%20-%2015%20x%20%2B%2035%20" alt="LaTeX:  \displaystyle 27 x^{3} - 63 x^{2} - 15 x + 35 " data-equation-content=" \displaystyle 27 x^{3} - 63 x^{2} - 15 x + 35 " /> . </p> </p>
HTML for Canvas
<p> <p>Grouping the first two terms and factoring out their GCF,  <img class="equation_image" title=" \displaystyle 9 x^{2} " src="/equation_images/%20%5Cdisplaystyle%209%20x%5E%7B2%7D%20" alt="LaTeX:  \displaystyle 9 x^{2} " data-equation-content=" \displaystyle 9 x^{2} " /> , gives  <img class="equation_image" title=" \displaystyle 9 x^{2}(3 x - 7) " src="/equation_images/%20%5Cdisplaystyle%209%20x%5E%7B2%7D%283%20x%20-%207%29%20" alt="LaTeX:  \displaystyle 9 x^{2}(3 x - 7) " data-equation-content=" \displaystyle 9 x^{2}(3 x - 7) " /> . Grouping the last two terms and factoring out their GCF,  <img class="equation_image" title=" \displaystyle -5 " src="/equation_images/%20%5Cdisplaystyle%20-5%20" alt="LaTeX:  \displaystyle -5 " data-equation-content=" \displaystyle -5 " /> , gives  <img class="equation_image" title=" \displaystyle -5(3 x - 7) " src="/equation_images/%20%5Cdisplaystyle%20-5%283%20x%20-%207%29%20" alt="LaTeX:  \displaystyle -5(3 x - 7) " data-equation-content=" \displaystyle -5(3 x - 7) " /> . The polynomial now has a common binomial factor of  <img class="equation_image" title=" \displaystyle 3 x - 7 " src="/equation_images/%20%5Cdisplaystyle%203%20x%20-%207%20" alt="LaTeX:  \displaystyle 3 x - 7 " data-equation-content=" \displaystyle 3 x - 7 " /> . This gives  <img class="equation_image" title=" \displaystyle 9 x^{2} \left(3 x - 7\right) -5 \cdot \left(3 x - 7\right) = \left(3 x - 7\right) \left(9 x^{2} - 5\right) " src="/equation_images/%20%5Cdisplaystyle%209%20x%5E%7B2%7D%20%5Cleft%283%20x%20-%207%5Cright%29%20-5%20%5Ccdot%20%5Cleft%283%20x%20-%207%5Cright%29%20%3D%20%5Cleft%283%20x%20-%207%5Cright%29%20%5Cleft%289%20x%5E%7B2%7D%20-%205%5Cright%29%20" alt="LaTeX:  \displaystyle 9 x^{2} \left(3 x - 7\right) -5 \cdot \left(3 x - 7\right) = \left(3 x - 7\right) \left(9 x^{2} - 5\right) " data-equation-content=" \displaystyle 9 x^{2} \left(3 x - 7\right) -5 \cdot \left(3 x - 7\right) = \left(3 x - 7\right) \left(9 x^{2} - 5\right) " /> . </p> </p>