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Factor \(\displaystyle 7 x^{3} - 9 x^{2} - 42 x + 54\).

Grouping the first two terms and factoring out their GCF, \(\displaystyle x^{2}\), gives \(\displaystyle x^{2}(7 x - 9)\). Grouping the last two terms and factoring out their GCF, \(\displaystyle -6\), gives \(\displaystyle -6(7 x - 9)\). The polynomial now has a common binomial factor of \(\displaystyle 7 x - 9\). This gives \(\displaystyle x^{2} \left(7 x - 9\right) -6 \cdot \left(7 x - 9\right) = \left(7 x - 9\right) \left(x^{2} - 6\right)\).

Download \(\LaTeX\) 

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

\end{question}

Download Question and Solution Environment\(\LaTeX\)
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HTML for Canvas 
<p> <p>Factor  <img class="equation_image" title=" \displaystyle 7 x^{3} - 9 x^{2} - 42 x + 54 " src="/equation_images/%20%5Cdisplaystyle%207%20x%5E%7B3%7D%20-%209%20x%5E%7B2%7D%20-%2042%20x%20%2B%2054%20" alt="LaTeX:  \displaystyle 7 x^{3} - 9 x^{2} - 42 x + 54 " data-equation-content=" \displaystyle 7 x^{3} - 9 x^{2} - 42 x + 54 " /> . </p> </p>
HTML for Canvas
<p> <p>Grouping the first two terms and factoring out their GCF,  <img class="equation_image" title=" \displaystyle x^{2} " src="/equation_images/%20%5Cdisplaystyle%20x%5E%7B2%7D%20" alt="LaTeX:  \displaystyle x^{2} " data-equation-content=" \displaystyle x^{2} " /> , gives  <img class="equation_image" title=" \displaystyle x^{2}(7 x - 9) " src="/equation_images/%20%5Cdisplaystyle%20x%5E%7B2%7D%287%20x%20-%209%29%20" alt="LaTeX:  \displaystyle x^{2}(7 x - 9) " data-equation-content=" \displaystyle x^{2}(7 x - 9) " /> . Grouping the last two terms and factoring out their GCF,  <img class="equation_image" title=" \displaystyle -6 " src="/equation_images/%20%5Cdisplaystyle%20-6%20" alt="LaTeX:  \displaystyle -6 " data-equation-content=" \displaystyle -6 " /> , gives  <img class="equation_image" title=" \displaystyle -6(7 x - 9) " src="/equation_images/%20%5Cdisplaystyle%20-6%287%20x%20-%209%29%20" alt="LaTeX:  \displaystyle -6(7 x - 9) " data-equation-content=" \displaystyle -6(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-%209%20" alt="LaTeX:  \displaystyle 7 x - 9 " data-equation-content=" \displaystyle 7 x - 9 " /> . This gives  <img class="equation_image" title=" \displaystyle x^{2} \left(7 x - 9\right) -6 \cdot \left(7 x - 9\right) = \left(7 x - 9\right) \left(x^{2} - 6\right) " src="/equation_images/%20%5Cdisplaystyle%20x%5E%7B2%7D%20%5Cleft%287%20x%20-%209%5Cright%29%20-6%20%5Ccdot%20%5Cleft%287%20x%20-%209%5Cright%29%20%3D%20%5Cleft%287%20x%20-%209%5Cright%29%20%5Cleft%28x%5E%7B2%7D%20-%206%5Cright%29%20" alt="LaTeX:  \displaystyle x^{2} \left(7 x - 9\right) -6 \cdot \left(7 x - 9\right) = \left(7 x - 9\right) \left(x^{2} - 6\right) " data-equation-content=" \displaystyle x^{2} \left(7 x - 9\right) -6 \cdot \left(7 x - 9\right) = \left(7 x - 9\right) \left(x^{2} - 6\right) " /> . </p> </p>