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Factoring out the GCF \(\displaystyle 4\) from each term gives \(\displaystyle 4(3 x^{3} - 2 x^{2} - 12 x + 8)\). Grouping the first two terms and factoring out their GCF, \(\displaystyle x^{2}\), gives \(\displaystyle x^{2}(3 x - 2)\). Grouping the last two terms and factoring out their GCF, \(\displaystyle -4\), gives \(\displaystyle -4(3 x - 2)\). The polynomial now has a common binomial factor of \(\displaystyle 3 x - 2\). This gives \(\displaystyle 4[x^{2} \left(3 x - 2\right) -4 \cdot \left(3 x - 2\right)] = 4\left(3 x - 2\right) \left(x^{2} - 4\right)\). The quadratic factor can be factored using the difference of squares to give \(\displaystyle 4\left(x - 2\right) \left(x + 2\right) \left(3 x - 2\right). \)

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\begin{question}Factor $12 x^{3} - 8 x^{2} - 48 x + 32$. 
    \soln{9cm}{Factoring out the GCF $4$ from each term gives $4(3 x^{3} - 2 x^{2} - 12 x + 8)$. Grouping the first two terms and factoring out their GCF, $x^{2}$, gives $x^{2}(3 x - 2)$. Grouping the last two terms and factoring out their GCF, $-4$, gives $-4(3 x - 2)$. The polynomial now has a common binomial factor of $3 x - 2$. This gives $4[x^{2} \left(3 x - 2\right) -4 \cdot \left(3 x - 2\right)] = 4\left(3 x - 2\right) \left(x^{2} - 4\right)$. The quadratic factor can be factored using the difference of squares to give $4\left(x - 2\right) \left(x + 2\right) \left(3 x - 2\right). $}

\end{question}

Download Question and Solution Environment\(\LaTeX\)

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\begin{document}\begin{question}(10pts) The question goes here!
    \soln{9cm}{The solution goes here.}

\end{question}\end{document}

HTML for Canvas

<p> <p>Factor  <img class="equation_image" title=" \displaystyle 12 x^{3} - 8 x^{2} - 48 x + 32 " src="/equation_images/%20%5Cdisplaystyle%2012%20x%5E%7B3%7D%20-%208%20x%5E%7B2%7D%20-%2048%20x%20%2B%2032%20" alt="LaTeX:  \displaystyle 12 x^{3} - 8 x^{2} - 48 x + 32 " data-equation-content=" \displaystyle 12 x^{3} - 8 x^{2} - 48 x + 32 " /> . </p> </p>

HTML for Canvas

<p> <p>Factoring out the GCF  <img class="equation_image" title=" \displaystyle 4 " src="/equation_images/%20%5Cdisplaystyle%204%20" alt="LaTeX:  \displaystyle 4 " data-equation-content=" \displaystyle 4 " />  from each term gives  <img class="equation_image" title=" \displaystyle 4(3 x^{3} - 2 x^{2} - 12 x + 8) " src="/equation_images/%20%5Cdisplaystyle%204%283%20x%5E%7B3%7D%20-%202%20x%5E%7B2%7D%20-%2012%20x%20%2B%208%29%20" alt="LaTeX:  \displaystyle 4(3 x^{3} - 2 x^{2} - 12 x + 8) " data-equation-content=" \displaystyle 4(3 x^{3} - 2 x^{2} - 12 x + 8) " /> . 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}(3 x - 2) " src="/equation_images/%20%5Cdisplaystyle%20x%5E%7B2%7D%283%20x%20-%202%29%20" alt="LaTeX:  \displaystyle x^{2}(3 x - 2) " data-equation-content=" \displaystyle x^{2}(3 x - 2) " /> . Grouping the last two terms and factoring out their GCF,  <img class="equation_image" title=" \displaystyle -4 " src="/equation_images/%20%5Cdisplaystyle%20-4%20" alt="LaTeX:  \displaystyle -4 " data-equation-content=" \displaystyle -4 " /> , gives  <img class="equation_image" title=" \displaystyle -4(3 x - 2) " src="/equation_images/%20%5Cdisplaystyle%20-4%283%20x%20-%202%29%20" alt="LaTeX:  \displaystyle -4(3 x - 2) " data-equation-content=" \displaystyle -4(3 x - 2) " /> . The polynomial now has a common binomial factor of  <img class="equation_image" title=" \displaystyle 3 x - 2 " src="/equation_images/%20%5Cdisplaystyle%203%20x%20-%202%20" alt="LaTeX:  \displaystyle 3 x - 2 " data-equation-content=" \displaystyle 3 x - 2 " /> . This gives  <img class="equation_image" title=" \displaystyle 4[x^{2} \left(3 x - 2\right) -4 \cdot \left(3 x - 2\right)] = 4\left(3 x - 2\right) \left(x^{2} - 4\right) " src="/equation_images/%20%5Cdisplaystyle%204%5Bx%5E%7B2%7D%20%5Cleft%283%20x%20-%202%5Cright%29%20-4%20%5Ccdot%20%5Cleft%283%20x%20-%202%5Cright%29%5D%20%3D%204%5Cleft%283%20x%20-%202%5Cright%29%20%5Cleft%28x%5E%7B2%7D%20-%204%5Cright%29%20" alt="LaTeX:  \displaystyle 4[x^{2} \left(3 x - 2\right) -4 \cdot \left(3 x - 2\right)] = 4\left(3 x - 2\right) \left(x^{2} - 4\right) " data-equation-content=" \displaystyle 4[x^{2} \left(3 x - 2\right) -4 \cdot \left(3 x - 2\right)] = 4\left(3 x - 2\right) \left(x^{2} - 4\right) " /> . The quadratic factor can be factored using the difference of squares to give  <img class="equation_image" title=" \displaystyle 4\left(x - 2\right) \left(x + 2\right) \left(3 x - 2\right).  " src="/equation_images/%20%5Cdisplaystyle%204%5Cleft%28x%20-%202%5Cright%29%20%5Cleft%28x%20%2B%202%5Cright%29%20%5Cleft%283%20x%20-%202%5Cright%29.%20%20" alt="LaTeX:  \displaystyle 4\left(x - 2\right) \left(x + 2\right) \left(3 x - 2\right).  " data-equation-content=" \displaystyle 4\left(x - 2\right) \left(x + 2\right) \left(3 x - 2\right).  " /> </p> </p>