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Factor \(\displaystyle - 36 x^{3} + 8 x^{2} - 54 x + 12\).

Factoring out the GCF \(\displaystyle -2\) from each term gives \(\displaystyle -2(18 x^{3} - 4 x^{2} + 27 x - 6)\). Grouping the first two terms and factoring out their GCF, \(\displaystyle 2 x^{2}\), gives \(\displaystyle 2 x^{2}(9 x - 2)\). Grouping the last two terms and factoring out their GCF, \(\displaystyle 3\), gives \(\displaystyle 3(9 x - 2)\). The polynomial now has a common binomial factor of \(\displaystyle 9 x - 2\). This gives \(\displaystyle -2[2 x^{2} \left(9 x - 2\right) +3 \cdot \left(9 x - 2\right)] = -2\left(9 x - 2\right) \left(2 x^{2} + 3\right)\).

Download \(\LaTeX\) 

\begin{question}Factor $- 36 x^{3} + 8 x^{2} - 54 x + 12$. 
    \soln{9cm}{Factoring out the GCF $-2$ from each term gives $-2(18 x^{3} - 4 x^{2} + 27 x - 6)$. Grouping the first two terms and factoring out their GCF, $2 x^{2}$, gives $2 x^{2}(9 x - 2)$. Grouping the last two terms and factoring out their GCF, $3$, gives $3(9 x - 2)$. The polynomial now has a common binomial factor of $9 x - 2$. This gives $-2[2 x^{2} \left(9 x - 2\right) +3 \cdot \left(9 x - 2\right)] = -2\left(9 x - 2\right) \left(2 x^{2} + 3\right)$. }

\end{question}

Download Question and Solution Environment\(\LaTeX\)
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HTML for Canvas 
<p> <p>Factor  <img class="equation_image" title=" \displaystyle - 36 x^{3} + 8 x^{2} - 54 x + 12 " src="/equation_images/%20%5Cdisplaystyle%20-%2036%20x%5E%7B3%7D%20%2B%208%20x%5E%7B2%7D%20-%2054%20x%20%2B%2012%20" alt="LaTeX:  \displaystyle - 36 x^{3} + 8 x^{2} - 54 x + 12 " data-equation-content=" \displaystyle - 36 x^{3} + 8 x^{2} - 54 x + 12 " /> . </p> </p>
HTML for Canvas
<p> <p>Factoring out the GCF  <img class="equation_image" title=" \displaystyle -2 " src="/equation_images/%20%5Cdisplaystyle%20-2%20" alt="LaTeX:  \displaystyle -2 " data-equation-content=" \displaystyle -2 " />  from each term gives  <img class="equation_image" title=" \displaystyle -2(18 x^{3} - 4 x^{2} + 27 x - 6) " src="/equation_images/%20%5Cdisplaystyle%20-2%2818%20x%5E%7B3%7D%20-%204%20x%5E%7B2%7D%20%2B%2027%20x%20-%206%29%20" alt="LaTeX:  \displaystyle -2(18 x^{3} - 4 x^{2} + 27 x - 6) " data-equation-content=" \displaystyle -2(18 x^{3} - 4 x^{2} + 27 x - 6) " /> . Grouping the first two terms and factoring out their GCF,  <img class="equation_image" title=" \displaystyle 2 x^{2} " src="/equation_images/%20%5Cdisplaystyle%202%20x%5E%7B2%7D%20" alt="LaTeX:  \displaystyle 2 x^{2} " data-equation-content=" \displaystyle 2 x^{2} " /> , gives  <img class="equation_image" title=" \displaystyle 2 x^{2}(9 x - 2) " src="/equation_images/%20%5Cdisplaystyle%202%20x%5E%7B2%7D%289%20x%20-%202%29%20" alt="LaTeX:  \displaystyle 2 x^{2}(9 x - 2) " data-equation-content=" \displaystyle 2 x^{2}(9 x - 2) " /> . Grouping the last two terms and factoring out their GCF,  <img class="equation_image" title=" \displaystyle 3 " src="/equation_images/%20%5Cdisplaystyle%203%20" alt="LaTeX:  \displaystyle 3 " data-equation-content=" \displaystyle 3 " /> , gives  <img class="equation_image" title=" \displaystyle 3(9 x - 2) " src="/equation_images/%20%5Cdisplaystyle%203%289%20x%20-%202%29%20" alt="LaTeX:  \displaystyle 3(9 x - 2) " data-equation-content=" \displaystyle 3(9 x - 2) " /> . The polynomial now has a common binomial factor of  <img class="equation_image" title=" \displaystyle 9 x - 2 " src="/equation_images/%20%5Cdisplaystyle%209%20x%20-%202%20" alt="LaTeX:  \displaystyle 9 x - 2 " data-equation-content=" \displaystyle 9 x - 2 " /> . This gives  <img class="equation_image" title=" \displaystyle -2[2 x^{2} \left(9 x - 2\right) +3 \cdot \left(9 x - 2\right)] = -2\left(9 x - 2\right) \left(2 x^{2} + 3\right) " src="/equation_images/%20%5Cdisplaystyle%20-2%5B2%20x%5E%7B2%7D%20%5Cleft%289%20x%20-%202%5Cright%29%20%2B3%20%5Ccdot%20%5Cleft%289%20x%20-%202%5Cright%29%5D%20%3D%20-2%5Cleft%289%20x%20-%202%5Cright%29%20%5Cleft%282%20x%5E%7B2%7D%20%2B%203%5Cright%29%20" alt="LaTeX:  \displaystyle -2[2 x^{2} \left(9 x - 2\right) +3 \cdot \left(9 x - 2\right)] = -2\left(9 x - 2\right) \left(2 x^{2} + 3\right) " data-equation-content=" \displaystyle -2[2 x^{2} \left(9 x - 2\right) +3 \cdot \left(9 x - 2\right)] = -2\left(9 x - 2\right) \left(2 x^{2} + 3\right) " /> . </p> </p>