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Factor \(\displaystyle - 18 x^{3} + 54 x^{2} + 3 x - 9\).

Factoring out the GCF \(\displaystyle -3\) from each term gives \(\displaystyle -3(6 x^{3} - 18 x^{2} - x + 3)\). Grouping the first two terms and factoring out their GCF, \(\displaystyle 6 x^{2}\), gives \(\displaystyle 6 x^{2}(x - 3)\). Grouping the last two terms and factoring out their GCF, \(\displaystyle -1\), gives \(\displaystyle -1(x - 3)\). The polynomial now has a common binomial factor of \(\displaystyle x - 3\). This gives \(\displaystyle -3[6 x^{2} \left(x - 3\right) -1 \cdot \left(x - 3\right)] = -3\left(x - 3\right) \left(6 x^{2} - 1\right)\).

Download \(\LaTeX\) 

\begin{question}Factor $- 18 x^{3} + 54 x^{2} + 3 x - 9$. 
    \soln{9cm}{Factoring out the GCF $-3$ from each term gives $-3(6 x^{3} - 18 x^{2} - x + 3)$. Grouping the first two terms and factoring out their GCF, $6 x^{2}$, gives $6 x^{2}(x - 3)$. Grouping the last two terms and factoring out their GCF, $-1$, gives $-1(x - 3)$. The polynomial now has a common binomial factor of $x - 3$. This gives $-3[6 x^{2} \left(x - 3\right) -1 \cdot \left(x - 3\right)] = -3\left(x - 3\right) \left(6 x^{2} - 1\right)$. }

\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} + 54 x^{2} + 3 x - 9 " src="/equation_images/%20%5Cdisplaystyle%20-%2018%20x%5E%7B3%7D%20%2B%2054%20x%5E%7B2%7D%20%2B%203%20x%20-%209%20" alt="LaTeX:  \displaystyle - 18 x^{3} + 54 x^{2} + 3 x - 9 " data-equation-content=" \displaystyle - 18 x^{3} + 54 x^{2} + 3 x - 9 " /> . </p> </p>
HTML for Canvas
<p> <p>Factoring out the 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 " />  from each term gives  <img class="equation_image" title=" \displaystyle -3(6 x^{3} - 18 x^{2} - x + 3) " src="/equation_images/%20%5Cdisplaystyle%20-3%286%20x%5E%7B3%7D%20-%2018%20x%5E%7B2%7D%20-%20x%20%2B%203%29%20" alt="LaTeX:  \displaystyle -3(6 x^{3} - 18 x^{2} - x + 3) " data-equation-content=" \displaystyle -3(6 x^{3} - 18 x^{2} - x + 3) " /> . Grouping the first two terms and factoring out their GCF,  <img class="equation_image" title=" \displaystyle 6 x^{2} " src="/equation_images/%20%5Cdisplaystyle%206%20x%5E%7B2%7D%20" alt="LaTeX:  \displaystyle 6 x^{2} " data-equation-content=" \displaystyle 6 x^{2} " /> , gives  <img class="equation_image" title=" \displaystyle 6 x^{2}(x - 3) " src="/equation_images/%20%5Cdisplaystyle%206%20x%5E%7B2%7D%28x%20-%203%29%20" alt="LaTeX:  \displaystyle 6 x^{2}(x - 3) " data-equation-content=" \displaystyle 6 x^{2}(x - 3) " /> . Grouping the last two terms and factoring out their GCF,  <img class="equation_image" title=" \displaystyle -1 " src="/equation_images/%20%5Cdisplaystyle%20-1%20" alt="LaTeX:  \displaystyle -1 " data-equation-content=" \displaystyle -1 " /> , gives  <img class="equation_image" title=" \displaystyle -1(x - 3) " src="/equation_images/%20%5Cdisplaystyle%20-1%28x%20-%203%29%20" alt="LaTeX:  \displaystyle -1(x - 3) " data-equation-content=" \displaystyle -1(x - 3) " /> . The polynomial now has a common binomial factor of  <img class="equation_image" title=" \displaystyle x - 3 " src="/equation_images/%20%5Cdisplaystyle%20x%20-%203%20" alt="LaTeX:  \displaystyle x - 3 " data-equation-content=" \displaystyle x - 3 " /> . This gives  <img class="equation_image" title=" \displaystyle -3[6 x^{2} \left(x - 3\right) -1 \cdot \left(x - 3\right)] = -3\left(x - 3\right) \left(6 x^{2} - 1\right) " src="/equation_images/%20%5Cdisplaystyle%20-3%5B6%20x%5E%7B2%7D%20%5Cleft%28x%20-%203%5Cright%29%20-1%20%5Ccdot%20%5Cleft%28x%20-%203%5Cright%29%5D%20%3D%20-3%5Cleft%28x%20-%203%5Cright%29%20%5Cleft%286%20x%5E%7B2%7D%20-%201%5Cright%29%20" alt="LaTeX:  \displaystyle -3[6 x^{2} \left(x - 3\right) -1 \cdot \left(x - 3\right)] = -3\left(x - 3\right) \left(6 x^{2} - 1\right) " data-equation-content=" \displaystyle -3[6 x^{2} \left(x - 3\right) -1 \cdot \left(x - 3\right)] = -3\left(x - 3\right) \left(6 x^{2} - 1\right) " /> . </p> </p>