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Algebra
Quadratics
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Factor \(\displaystyle 2 x^{3} - 9 x^{2} - 10 x + 45\).

Grouping the first two terms and factoring out their GCF, \(\displaystyle x^{2}\), gives \(\displaystyle x^{2}(2 x - 9)\). Grouping the last two terms and factoring out their GCF, \(\displaystyle -5\), gives \(\displaystyle -5(2 x - 9)\). The polynomial now has a common binomial factor of \(\displaystyle 2 x - 9\). This gives \(\displaystyle x^{2} \left(2 x - 9\right) -5 \cdot \left(2 x - 9\right) = \left(2 x - 9\right) \left(x^{2} - 5\right)\).

Download \(\LaTeX\) 

\begin{question}Factor $2 x^{3} - 9 x^{2} - 10 x + 45$. 
    \soln{9cm}{Grouping the first two terms and factoring out their GCF, $x^{2}$, gives $x^{2}(2 x - 9)$. Grouping the last two terms and factoring out their GCF, $-5$, gives $-5(2 x - 9)$. The polynomial now has a common binomial factor of $2 x - 9$. This gives $x^{2} \left(2 x - 9\right) -5 \cdot \left(2 x - 9\right) = \left(2 x - 9\right) \left(x^{2} - 5\right)$. }

\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} - 9 x^{2} - 10 x + 45 " src="/equation_images/%20%5Cdisplaystyle%202%20x%5E%7B3%7D%20-%209%20x%5E%7B2%7D%20-%2010%20x%20%2B%2045%20" alt="LaTeX:  \displaystyle 2 x^{3} - 9 x^{2} - 10 x + 45 " data-equation-content=" \displaystyle 2 x^{3} - 9 x^{2} - 10 x + 45 " /> . </p> </p>
HTML for Canvas
<p> <p>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 - 9) " src="/equation_images/%20%5Cdisplaystyle%20x%5E%7B2%7D%282%20x%20-%209%29%20" alt="LaTeX:  \displaystyle x^{2}(2 x - 9) " data-equation-content=" \displaystyle x^{2}(2 x - 9) " /> . Grouping the last two terms and factoring out their GCF,  <img class="equation_image" title=" \displaystyle -5 " src="/equation_images/%20%5Cdisplaystyle%20-5%20" alt="LaTeX:  \displaystyle -5 " data-equation-content=" \displaystyle -5 " /> , gives  <img class="equation_image" title=" \displaystyle -5(2 x - 9) " src="/equation_images/%20%5Cdisplaystyle%20-5%282%20x%20-%209%29%20" alt="LaTeX:  \displaystyle -5(2 x - 9) " data-equation-content=" \displaystyle -5(2 x - 9) " /> . The polynomial now has a common binomial factor of  <img class="equation_image" title=" \displaystyle 2 x - 9 " src="/equation_images/%20%5Cdisplaystyle%202%20x%20-%209%20" alt="LaTeX:  \displaystyle 2 x - 9 " data-equation-content=" \displaystyle 2 x - 9 " /> . This gives  <img class="equation_image" title=" \displaystyle x^{2} \left(2 x - 9\right) -5 \cdot \left(2 x - 9\right) = \left(2 x - 9\right) \left(x^{2} - 5\right) " src="/equation_images/%20%5Cdisplaystyle%20x%5E%7B2%7D%20%5Cleft%282%20x%20-%209%5Cright%29%20-5%20%5Ccdot%20%5Cleft%282%20x%20-%209%5Cright%29%20%3D%20%5Cleft%282%20x%20-%209%5Cright%29%20%5Cleft%28x%5E%7B2%7D%20-%205%5Cright%29%20" alt="LaTeX:  \displaystyle x^{2} \left(2 x - 9\right) -5 \cdot \left(2 x - 9\right) = \left(2 x - 9\right) \left(x^{2} - 5\right) " data-equation-content=" \displaystyle x^{2} \left(2 x - 9\right) -5 \cdot \left(2 x - 9\right) = \left(2 x - 9\right) \left(x^{2} - 5\right) " /> . </p> </p>