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Factor \(\displaystyle 90 x^{3} - 40 x^{2} - 54 x + 24\).

Factoring out the GCF \(\displaystyle 2\) from each term gives \(\displaystyle 2(45 x^{3} - 20 x^{2} - 27 x + 12)\). Grouping the first two terms and factoring out their GCF, \(\displaystyle 5 x^{2}\), gives \(\displaystyle 5 x^{2}(9 x - 4)\). Grouping the last two terms and factoring out their GCF, \(\displaystyle -3\), gives \(\displaystyle -3(9 x - 4)\). The polynomial now has a common binomial factor of \(\displaystyle 9 x - 4\). This gives \(\displaystyle 2[5 x^{2} \left(9 x - 4\right) -3 \cdot \left(9 x - 4\right)] = 2\left(9 x - 4\right) \left(5 x^{2} - 3\right)\).

Download \(\LaTeX\) 

\begin{question}Factor $90 x^{3} - 40 x^{2} - 54 x + 24$. 
    \soln{9cm}{Factoring out the GCF $2$ from each term gives $2(45 x^{3} - 20 x^{2} - 27 x + 12)$. Grouping the first two terms and factoring out their GCF, $5 x^{2}$, gives $5 x^{2}(9 x - 4)$. Grouping the last two terms and factoring out their GCF, $-3$, gives $-3(9 x - 4)$. The polynomial now has a common binomial factor of $9 x - 4$. This gives $2[5 x^{2} \left(9 x - 4\right) -3 \cdot \left(9 x - 4\right)] = 2\left(9 x - 4\right) \left(5 x^{2} - 3\right)$. }

\end{question}

Download Question and Solution Environment\(\LaTeX\)
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HTML for Canvas 
<p> <p>Factor  <img class="equation_image" title=" \displaystyle 90 x^{3} - 40 x^{2} - 54 x + 24 " src="/equation_images/%20%5Cdisplaystyle%2090%20x%5E%7B3%7D%20-%2040%20x%5E%7B2%7D%20-%2054%20x%20%2B%2024%20" alt="LaTeX:  \displaystyle 90 x^{3} - 40 x^{2} - 54 x + 24 " data-equation-content=" \displaystyle 90 x^{3} - 40 x^{2} - 54 x + 24 " /> . </p> </p>
HTML for Canvas
<p> <p>Factoring out the GCF  <img class="equation_image" title=" \displaystyle 2 " src="/equation_images/%20%5Cdisplaystyle%202%20" alt="LaTeX:  \displaystyle 2 " data-equation-content=" \displaystyle 2 " />  from each term gives  <img class="equation_image" title=" \displaystyle 2(45 x^{3} - 20 x^{2} - 27 x + 12) " src="/equation_images/%20%5Cdisplaystyle%202%2845%20x%5E%7B3%7D%20-%2020%20x%5E%7B2%7D%20-%2027%20x%20%2B%2012%29%20" alt="LaTeX:  \displaystyle 2(45 x^{3} - 20 x^{2} - 27 x + 12) " data-equation-content=" \displaystyle 2(45 x^{3} - 20 x^{2} - 27 x + 12) " /> . Grouping the first two terms and factoring out their GCF,  <img class="equation_image" title=" \displaystyle 5 x^{2} " src="/equation_images/%20%5Cdisplaystyle%205%20x%5E%7B2%7D%20" alt="LaTeX:  \displaystyle 5 x^{2} " data-equation-content=" \displaystyle 5 x^{2} " /> , gives  <img class="equation_image" title=" \displaystyle 5 x^{2}(9 x - 4) " src="/equation_images/%20%5Cdisplaystyle%205%20x%5E%7B2%7D%289%20x%20-%204%29%20" alt="LaTeX:  \displaystyle 5 x^{2}(9 x - 4) " data-equation-content=" \displaystyle 5 x^{2}(9 x - 4) " /> . Grouping the last two terms and factoring out their 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 " /> , gives  <img class="equation_image" title=" \displaystyle -3(9 x - 4) " src="/equation_images/%20%5Cdisplaystyle%20-3%289%20x%20-%204%29%20" alt="LaTeX:  \displaystyle -3(9 x - 4) " data-equation-content=" \displaystyle -3(9 x - 4) " /> . The polynomial now has a common binomial factor of  <img class="equation_image" title=" \displaystyle 9 x - 4 " src="/equation_images/%20%5Cdisplaystyle%209%20x%20-%204%20" alt="LaTeX:  \displaystyle 9 x - 4 " data-equation-content=" \displaystyle 9 x - 4 " /> . This gives  <img class="equation_image" title=" \displaystyle 2[5 x^{2} \left(9 x - 4\right) -3 \cdot \left(9 x - 4\right)] = 2\left(9 x - 4\right) \left(5 x^{2} - 3\right) " src="/equation_images/%20%5Cdisplaystyle%202%5B5%20x%5E%7B2%7D%20%5Cleft%289%20x%20-%204%5Cright%29%20-3%20%5Ccdot%20%5Cleft%289%20x%20-%204%5Cright%29%5D%20%3D%202%5Cleft%289%20x%20-%204%5Cright%29%20%5Cleft%285%20x%5E%7B2%7D%20-%203%5Cright%29%20" alt="LaTeX:  \displaystyle 2[5 x^{2} \left(9 x - 4\right) -3 \cdot \left(9 x - 4\right)] = 2\left(9 x - 4\right) \left(5 x^{2} - 3\right) " data-equation-content=" \displaystyle 2[5 x^{2} \left(9 x - 4\right) -3 \cdot \left(9 x - 4\right)] = 2\left(9 x - 4\right) \left(5 x^{2} - 3\right) " /> . </p> </p>