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Algebra
Quadratics
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Factor \(\displaystyle 18 x^{3} - 63 x^{2} - 8 x + 28\).

Grouping the first two terms and factoring out their GCF, \(\displaystyle 9 x^{2}\), gives \(\displaystyle 9 x^{2}(2 x - 7)\). Grouping the last two terms and factoring out their GCF, \(\displaystyle -4\), gives \(\displaystyle -4(2 x - 7)\). The polynomial now has a common binomial factor of \(\displaystyle 2 x - 7\). This gives \(\displaystyle 9 x^{2} \left(2 x - 7\right) -4 \cdot \left(2 x - 7\right) = \left(2 x - 7\right) \left(9 x^{2} - 4\right)\). The quadratic factor can be factored using the difference of squares to give \(\displaystyle \left(2 x - 7\right) \left(3 x - 2\right) \left(3 x + 2\right). \)

Download \(\LaTeX\) 

\begin{question}Factor $18 x^{3} - 63 x^{2} - 8 x + 28$. 
    \soln{9cm}{Grouping the first two terms and factoring out their GCF, $9 x^{2}$, gives $9 x^{2}(2 x - 7)$. Grouping the last two terms and factoring out their GCF, $-4$, gives $-4(2 x - 7)$. The polynomial now has a common binomial factor of $2 x - 7$. This gives $9 x^{2} \left(2 x - 7\right) -4 \cdot \left(2 x - 7\right) = \left(2 x - 7\right) \left(9 x^{2} - 4\right)$. The quadratic factor can be factored using the difference of squares to give $\left(2 x - 7\right) \left(3 x - 2\right) \left(3 x + 2\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} - 63 x^{2} - 8 x + 28 " src="/equation_images/%20%5Cdisplaystyle%2018%20x%5E%7B3%7D%20-%2063%20x%5E%7B2%7D%20-%208%20x%20%2B%2028%20" alt="LaTeX:  \displaystyle 18 x^{3} - 63 x^{2} - 8 x + 28 " data-equation-content=" \displaystyle 18 x^{3} - 63 x^{2} - 8 x + 28 " /> . </p> </p>
HTML for Canvas
<p> <p>Grouping the first two terms and factoring out their GCF,  <img class="equation_image" title=" \displaystyle 9 x^{2} " src="/equation_images/%20%5Cdisplaystyle%209%20x%5E%7B2%7D%20" alt="LaTeX:  \displaystyle 9 x^{2} " data-equation-content=" \displaystyle 9 x^{2} " /> , gives  <img class="equation_image" title=" \displaystyle 9 x^{2}(2 x - 7) " src="/equation_images/%20%5Cdisplaystyle%209%20x%5E%7B2%7D%282%20x%20-%207%29%20" alt="LaTeX:  \displaystyle 9 x^{2}(2 x - 7) " data-equation-content=" \displaystyle 9 x^{2}(2 x - 7) " /> . 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(2 x - 7) " src="/equation_images/%20%5Cdisplaystyle%20-4%282%20x%20-%207%29%20" alt="LaTeX:  \displaystyle -4(2 x - 7) " data-equation-content=" \displaystyle -4(2 x - 7) " /> . The polynomial now has a common binomial factor of  <img class="equation_image" title=" \displaystyle 2 x - 7 " src="/equation_images/%20%5Cdisplaystyle%202%20x%20-%207%20" alt="LaTeX:  \displaystyle 2 x - 7 " data-equation-content=" \displaystyle 2 x - 7 " /> . This gives  <img class="equation_image" title=" \displaystyle 9 x^{2} \left(2 x - 7\right) -4 \cdot \left(2 x - 7\right) = \left(2 x - 7\right) \left(9 x^{2} - 4\right) " src="/equation_images/%20%5Cdisplaystyle%209%20x%5E%7B2%7D%20%5Cleft%282%20x%20-%207%5Cright%29%20-4%20%5Ccdot%20%5Cleft%282%20x%20-%207%5Cright%29%20%3D%20%5Cleft%282%20x%20-%207%5Cright%29%20%5Cleft%289%20x%5E%7B2%7D%20-%204%5Cright%29%20" alt="LaTeX:  \displaystyle 9 x^{2} \left(2 x - 7\right) -4 \cdot \left(2 x - 7\right) = \left(2 x - 7\right) \left(9 x^{2} - 4\right) " data-equation-content=" \displaystyle 9 x^{2} \left(2 x - 7\right) -4 \cdot \left(2 x - 7\right) = \left(2 x - 7\right) \left(9 x^{2} - 4\right) " /> . The quadratic factor can be factored using the difference of squares to give  <img class="equation_image" title=" \displaystyle \left(2 x - 7\right) \left(3 x - 2\right) \left(3 x + 2\right).  " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%282%20x%20-%207%5Cright%29%20%5Cleft%283%20x%20-%202%5Cright%29%20%5Cleft%283%20x%20%2B%202%5Cright%29.%20%20" alt="LaTeX:  \displaystyle \left(2 x - 7\right) \left(3 x - 2\right) \left(3 x + 2\right).  " data-equation-content=" \displaystyle \left(2 x - 7\right) \left(3 x - 2\right) \left(3 x + 2\right).  " /> </p> </p>