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Factor \(\displaystyle 8 x^{3} - 4 x^{2} - 48 x + 24\).

Factoring out the GCF \(\displaystyle 4\) from each term gives \(\displaystyle 4(2 x^{3} - x^{2} - 12 x + 6)\). Grouping the first two terms and factoring out their GCF, \(\displaystyle x^{2}\), gives \(\displaystyle x^{2}(2 x - 1)\). Grouping the last two terms and factoring out their GCF, \(\displaystyle -6\), gives \(\displaystyle -6(2 x - 1)\). The polynomial now has a common binomial factor of \(\displaystyle 2 x - 1\). This gives \(\displaystyle 4[x^{2} \left(2 x - 1\right) -6 \cdot \left(2 x - 1\right)] = 4\left(2 x - 1\right) \left(x^{2} - 6\right)\).

Download \(\LaTeX\) 

\begin{question}Factor $8 x^{3} - 4 x^{2} - 48 x + 24$. 
    \soln{9cm}{Factoring out the GCF $4$ from each term gives $4(2 x^{3} - x^{2} - 12 x + 6)$. Grouping the first two terms and factoring out their GCF, $x^{2}$, gives $x^{2}(2 x - 1)$. Grouping the last two terms and factoring out their GCF, $-6$, gives $-6(2 x - 1)$. The polynomial now has a common binomial factor of $2 x - 1$. This gives $4[x^{2} \left(2 x - 1\right) -6 \cdot \left(2 x - 1\right)] = 4\left(2 x - 1\right) \left(x^{2} - 6\right)$. }

\end{question}

Download Question and Solution Environment\(\LaTeX\)
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HTML for Canvas 
<p> <p>Factor  <img class="equation_image" title=" \displaystyle 8 x^{3} - 4 x^{2} - 48 x + 24 " src="/equation_images/%20%5Cdisplaystyle%208%20x%5E%7B3%7D%20-%204%20x%5E%7B2%7D%20-%2048%20x%20%2B%2024%20" alt="LaTeX:  \displaystyle 8 x^{3} - 4 x^{2} - 48 x + 24 " data-equation-content=" \displaystyle 8 x^{3} - 4 x^{2} - 48 x + 24 " /> . </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(2 x^{3} - x^{2} - 12 x + 6) " src="/equation_images/%20%5Cdisplaystyle%204%282%20x%5E%7B3%7D%20-%20x%5E%7B2%7D%20-%2012%20x%20%2B%206%29%20" alt="LaTeX:  \displaystyle 4(2 x^{3} - x^{2} - 12 x + 6) " data-equation-content=" \displaystyle 4(2 x^{3} - x^{2} - 12 x + 6) " /> . 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}(2 x - 1) " src="/equation_images/%20%5Cdisplaystyle%20x%5E%7B2%7D%282%20x%20-%201%29%20" alt="LaTeX:  \displaystyle x^{2}(2 x - 1) " data-equation-content=" \displaystyle x^{2}(2 x - 1) " /> . 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(2 x - 1) " src="/equation_images/%20%5Cdisplaystyle%20-6%282%20x%20-%201%29%20" alt="LaTeX:  \displaystyle -6(2 x - 1) " data-equation-content=" \displaystyle -6(2 x - 1) " /> . The polynomial now has a common binomial factor of  <img class="equation_image" title=" \displaystyle 2 x - 1 " src="/equation_images/%20%5Cdisplaystyle%202%20x%20-%201%20" alt="LaTeX:  \displaystyle 2 x - 1 " data-equation-content=" \displaystyle 2 x - 1 " /> . This gives  <img class="equation_image" title=" \displaystyle 4[x^{2} \left(2 x - 1\right) -6 \cdot \left(2 x - 1\right)] = 4\left(2 x - 1\right) \left(x^{2} - 6\right) " src="/equation_images/%20%5Cdisplaystyle%204%5Bx%5E%7B2%7D%20%5Cleft%282%20x%20-%201%5Cright%29%20-6%20%5Ccdot%20%5Cleft%282%20x%20-%201%5Cright%29%5D%20%3D%204%5Cleft%282%20x%20-%201%5Cright%29%20%5Cleft%28x%5E%7B2%7D%20-%206%5Cright%29%20" alt="LaTeX:  \displaystyle 4[x^{2} \left(2 x - 1\right) -6 \cdot \left(2 x - 1\right)] = 4\left(2 x - 1\right) \left(x^{2} - 6\right) " data-equation-content=" \displaystyle 4[x^{2} \left(2 x - 1\right) -6 \cdot \left(2 x - 1\right)] = 4\left(2 x - 1\right) \left(x^{2} - 6\right) " /> . </p> </p>