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Quadratics
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Factor \(\displaystyle - 2 x^{3} + 8 x^{2} + 18 x - 72\).

Factoring out the GCF \(\displaystyle -2\) from each term gives \(\displaystyle -2(x^{3} - 4 x^{2} - 9 x + 36)\). Grouping the first two terms and factoring out their GCF, \(\displaystyle x^{2}\), gives \(\displaystyle x^{2}(x - 4)\). Grouping the last two terms and factoring out their GCF, \(\displaystyle -9\), gives \(\displaystyle -9(x - 4)\). The polynomial now has a common binomial factor of \(\displaystyle x - 4\). This gives \(\displaystyle -2[x^{2} \left(x - 4\right) -9 \cdot \left(x - 4\right)] = -2\left(x - 4\right) \left(x^{2} - 9\right)\). The quadratic factor can be factored using the difference of squares to give \(\displaystyle -2\left(x - 4\right) \left(x - 3\right) \left(x + 3\right). \)

Download \(\LaTeX\) 

\begin{question}Factor $- 2 x^{3} + 8 x^{2} + 18 x - 72$. 
    \soln{9cm}{Factoring out the GCF $-2$ from each term gives $-2(x^{3} - 4 x^{2} - 9 x + 36)$. Grouping the first two terms and factoring out their GCF, $x^{2}$, gives $x^{2}(x - 4)$. Grouping the last two terms and factoring out their GCF, $-9$, gives $-9(x - 4)$. The polynomial now has a common binomial factor of $x - 4$. This gives $-2[x^{2} \left(x - 4\right) -9 \cdot \left(x - 4\right)] = -2\left(x - 4\right) \left(x^{2} - 9\right)$. The quadratic factor can be factored using the difference of squares to give $-2\left(x - 4\right) \left(x - 3\right) \left(x + 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 - 2 x^{3} + 8 x^{2} + 18 x - 72 " src="/equation_images/%20%5Cdisplaystyle%20-%202%20x%5E%7B3%7D%20%2B%208%20x%5E%7B2%7D%20%2B%2018%20x%20-%2072%20" alt="LaTeX:  \displaystyle - 2 x^{3} + 8 x^{2} + 18 x - 72 " data-equation-content=" \displaystyle - 2 x^{3} + 8 x^{2} + 18 x - 72 " /> . </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(x^{3} - 4 x^{2} - 9 x + 36) " src="/equation_images/%20%5Cdisplaystyle%20-2%28x%5E%7B3%7D%20-%204%20x%5E%7B2%7D%20-%209%20x%20%2B%2036%29%20" alt="LaTeX:  \displaystyle -2(x^{3} - 4 x^{2} - 9 x + 36) " data-equation-content=" \displaystyle -2(x^{3} - 4 x^{2} - 9 x + 36) " /> . 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}(x - 4) " src="/equation_images/%20%5Cdisplaystyle%20x%5E%7B2%7D%28x%20-%204%29%20" alt="LaTeX:  \displaystyle x^{2}(x - 4) " data-equation-content=" \displaystyle x^{2}(x - 4) " /> . Grouping the last two terms and factoring out their GCF,  <img class="equation_image" title=" \displaystyle -9 " src="/equation_images/%20%5Cdisplaystyle%20-9%20" alt="LaTeX:  \displaystyle -9 " data-equation-content=" \displaystyle -9 " /> , gives  <img class="equation_image" title=" \displaystyle -9(x - 4) " src="/equation_images/%20%5Cdisplaystyle%20-9%28x%20-%204%29%20" alt="LaTeX:  \displaystyle -9(x - 4) " data-equation-content=" \displaystyle -9(x - 4) " /> . The polynomial now has a common binomial factor of  <img class="equation_image" title=" \displaystyle x - 4 " src="/equation_images/%20%5Cdisplaystyle%20x%20-%204%20" alt="LaTeX:  \displaystyle x - 4 " data-equation-content=" \displaystyle x - 4 " /> . This gives  <img class="equation_image" title=" \displaystyle -2[x^{2} \left(x - 4\right) -9 \cdot \left(x - 4\right)] = -2\left(x - 4\right) \left(x^{2} - 9\right) " src="/equation_images/%20%5Cdisplaystyle%20-2%5Bx%5E%7B2%7D%20%5Cleft%28x%20-%204%5Cright%29%20-9%20%5Ccdot%20%5Cleft%28x%20-%204%5Cright%29%5D%20%3D%20-2%5Cleft%28x%20-%204%5Cright%29%20%5Cleft%28x%5E%7B2%7D%20-%209%5Cright%29%20" alt="LaTeX:  \displaystyle -2[x^{2} \left(x - 4\right) -9 \cdot \left(x - 4\right)] = -2\left(x - 4\right) \left(x^{2} - 9\right) " data-equation-content=" \displaystyle -2[x^{2} \left(x - 4\right) -9 \cdot \left(x - 4\right)] = -2\left(x - 4\right) \left(x^{2} - 9\right) " /> . The quadratic factor can be factored using the difference of squares to give  <img class="equation_image" title=" \displaystyle -2\left(x - 4\right) \left(x - 3\right) \left(x + 3\right).  " src="/equation_images/%20%5Cdisplaystyle%20-2%5Cleft%28x%20-%204%5Cright%29%20%5Cleft%28x%20-%203%5Cright%29%20%5Cleft%28x%20%2B%203%5Cright%29.%20%20" alt="LaTeX:  \displaystyle -2\left(x - 4\right) \left(x - 3\right) \left(x + 3\right).  " data-equation-content=" \displaystyle -2\left(x - 4\right) \left(x - 3\right) \left(x + 3\right).  " /> </p> </p>