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Factor \(\displaystyle 18 x^{3} - 81 x^{2} - 16 x + 72\).

Grouping the first two terms and factoring out their GCF, \(\displaystyle 9 x^{2}\), gives \(\displaystyle 9 x^{2}(2 x - 9)\). Grouping the last two terms and factoring out their GCF, \(\displaystyle -8\), gives \(\displaystyle -8(2 x - 9)\). The polynomial now has a common binomial factor of \(\displaystyle 2 x - 9\). This gives \(\displaystyle 9 x^{2} \left(2 x - 9\right) -8 \cdot \left(2 x - 9\right) = \left(2 x - 9\right) \left(9 x^{2} - 8\right)\).

Download \(\LaTeX\) 

\begin{question}Factor $18 x^{3} - 81 x^{2} - 16 x + 72$. 
    \soln{9cm}{Grouping the first two terms and factoring out their GCF, $9 x^{2}$, gives $9 x^{2}(2 x - 9)$. Grouping the last two terms and factoring out their GCF, $-8$, gives $-8(2 x - 9)$. The polynomial now has a common binomial factor of $2 x - 9$. This gives $9 x^{2} \left(2 x - 9\right) -8 \cdot \left(2 x - 9\right) = \left(2 x - 9\right) \left(9 x^{2} - 8\right)$. }

\end{question}

Download Question and Solution Environment\(\LaTeX\)
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HTML for Canvas 
<p> <p>Factor  <img class="equation_image" title=" \displaystyle 18 x^{3} - 81 x^{2} - 16 x + 72 " src="/equation_images/%20%5Cdisplaystyle%2018%20x%5E%7B3%7D%20-%2081%20x%5E%7B2%7D%20-%2016%20x%20%2B%2072%20" alt="LaTeX:  \displaystyle 18 x^{3} - 81 x^{2} - 16 x + 72 " data-equation-content=" \displaystyle 18 x^{3} - 81 x^{2} - 16 x + 72 " /> . </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}(2 x - 9) " src="/equation_images/%20%5Cdisplaystyle%209%20x%5E%7B2%7D%282%20x%20-%209%29%20" alt="LaTeX:  \displaystyle 9 x^{2}(2 x - 9) " data-equation-content=" \displaystyle 9 x^{2}(2 x - 9) " /> . Grouping the last two terms and factoring out their GCF,  <img class="equation_image" title=" \displaystyle -8 " src="/equation_images/%20%5Cdisplaystyle%20-8%20" alt="LaTeX:  \displaystyle -8 " data-equation-content=" \displaystyle -8 " /> , gives  <img class="equation_image" title=" \displaystyle -8(2 x - 9) " src="/equation_images/%20%5Cdisplaystyle%20-8%282%20x%20-%209%29%20" alt="LaTeX:  \displaystyle -8(2 x - 9) " data-equation-content=" \displaystyle -8(2 x - 9) " /> . The polynomial now has a common binomial factor of  <img class="equation_image" title=" \displaystyle 2 x - 9 " src="/equation_images/%20%5Cdisplaystyle%202%20x%20-%209%20" alt="LaTeX:  \displaystyle 2 x - 9 " data-equation-content=" \displaystyle 2 x - 9 " /> . This gives  <img class="equation_image" title=" \displaystyle 9 x^{2} \left(2 x - 9\right) -8 \cdot \left(2 x - 9\right) = \left(2 x - 9\right) \left(9 x^{2} - 8\right) " src="/equation_images/%20%5Cdisplaystyle%209%20x%5E%7B2%7D%20%5Cleft%282%20x%20-%209%5Cright%29%20-8%20%5Ccdot%20%5Cleft%282%20x%20-%209%5Cright%29%20%3D%20%5Cleft%282%20x%20-%209%5Cright%29%20%5Cleft%289%20x%5E%7B2%7D%20-%208%5Cright%29%20" alt="LaTeX:  \displaystyle 9 x^{2} \left(2 x - 9\right) -8 \cdot \left(2 x - 9\right) = \left(2 x - 9\right) \left(9 x^{2} - 8\right) " data-equation-content=" \displaystyle 9 x^{2} \left(2 x - 9\right) -8 \cdot \left(2 x - 9\right) = \left(2 x - 9\right) \left(9 x^{2} - 8\right) " /> . </p> </p>