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Factor \(\displaystyle - 60 x^{3} + 18 x^{2} + 90 x - 27\).

Factoring out the GCF \(\displaystyle -3\) from each term gives \(\displaystyle -3(20 x^{3} - 6 x^{2} - 30 x + 9)\). Grouping the first two terms and factoring out their GCF, \(\displaystyle 2 x^{2}\), gives \(\displaystyle 2 x^{2}(10 x - 3)\). Grouping the last two terms and factoring out their GCF, \(\displaystyle -3\), gives \(\displaystyle -3(10 x - 3)\). The polynomial now has a common binomial factor of \(\displaystyle 10 x - 3\). This gives \(\displaystyle -3[2 x^{2} \left(10 x - 3\right) -3 \cdot \left(10 x - 3\right)] = -3\left(10 x - 3\right) \left(2 x^{2} - 3\right)\).

Download \(\LaTeX\) 

\begin{question}Factor $- 60 x^{3} + 18 x^{2} + 90 x - 27$. 
    \soln{9cm}{Factoring out the GCF $-3$ from each term gives $-3(20 x^{3} - 6 x^{2} - 30 x + 9)$. Grouping the first two terms and factoring out their GCF, $2 x^{2}$, gives $2 x^{2}(10 x - 3)$. Grouping the last two terms and factoring out their GCF, $-3$, gives $-3(10 x - 3)$. The polynomial now has a common binomial factor of $10 x - 3$. This gives $-3[2 x^{2} \left(10 x - 3\right) -3 \cdot \left(10 x - 3\right)] = -3\left(10 x - 3\right) \left(2 x^{2} - 3\right)$. }

\end{question}

Download Question and Solution Environment\(\LaTeX\)
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HTML for Canvas 
<p> <p>Factor  <img class="equation_image" title=" \displaystyle - 60 x^{3} + 18 x^{2} + 90 x - 27 " src="/equation_images/%20%5Cdisplaystyle%20-%2060%20x%5E%7B3%7D%20%2B%2018%20x%5E%7B2%7D%20%2B%2090%20x%20-%2027%20" alt="LaTeX:  \displaystyle - 60 x^{3} + 18 x^{2} + 90 x - 27 " data-equation-content=" \displaystyle - 60 x^{3} + 18 x^{2} + 90 x - 27 " /> . </p> </p>
HTML for Canvas
<p> <p>Factoring out the 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 " />  from each term gives  <img class="equation_image" title=" \displaystyle -3(20 x^{3} - 6 x^{2} - 30 x + 9) " src="/equation_images/%20%5Cdisplaystyle%20-3%2820%20x%5E%7B3%7D%20-%206%20x%5E%7B2%7D%20-%2030%20x%20%2B%209%29%20" alt="LaTeX:  \displaystyle -3(20 x^{3} - 6 x^{2} - 30 x + 9) " data-equation-content=" \displaystyle -3(20 x^{3} - 6 x^{2} - 30 x + 9) " /> . Grouping the first two terms and factoring out their GCF,  <img class="equation_image" title=" \displaystyle 2 x^{2} " src="/equation_images/%20%5Cdisplaystyle%202%20x%5E%7B2%7D%20" alt="LaTeX:  \displaystyle 2 x^{2} " data-equation-content=" \displaystyle 2 x^{2} " /> , gives  <img class="equation_image" title=" \displaystyle 2 x^{2}(10 x - 3) " src="/equation_images/%20%5Cdisplaystyle%202%20x%5E%7B2%7D%2810%20x%20-%203%29%20" alt="LaTeX:  \displaystyle 2 x^{2}(10 x - 3) " data-equation-content=" \displaystyle 2 x^{2}(10 x - 3) " /> . 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(10 x - 3) " src="/equation_images/%20%5Cdisplaystyle%20-3%2810%20x%20-%203%29%20" alt="LaTeX:  \displaystyle -3(10 x - 3) " data-equation-content=" \displaystyle -3(10 x - 3) " /> . The polynomial now has a common binomial factor of  <img class="equation_image" title=" \displaystyle 10 x - 3 " src="/equation_images/%20%5Cdisplaystyle%2010%20x%20-%203%20" alt="LaTeX:  \displaystyle 10 x - 3 " data-equation-content=" \displaystyle 10 x - 3 " /> . This gives  <img class="equation_image" title=" \displaystyle -3[2 x^{2} \left(10 x - 3\right) -3 \cdot \left(10 x - 3\right)] = -3\left(10 x - 3\right) \left(2 x^{2} - 3\right) " src="/equation_images/%20%5Cdisplaystyle%20-3%5B2%20x%5E%7B2%7D%20%5Cleft%2810%20x%20-%203%5Cright%29%20-3%20%5Ccdot%20%5Cleft%2810%20x%20-%203%5Cright%29%5D%20%3D%20-3%5Cleft%2810%20x%20-%203%5Cright%29%20%5Cleft%282%20x%5E%7B2%7D%20-%203%5Cright%29%20" alt="LaTeX:  \displaystyle -3[2 x^{2} \left(10 x - 3\right) -3 \cdot \left(10 x - 3\right)] = -3\left(10 x - 3\right) \left(2 x^{2} - 3\right) " data-equation-content=" \displaystyle -3[2 x^{2} \left(10 x - 3\right) -3 \cdot \left(10 x - 3\right)] = -3\left(10 x - 3\right) \left(2 x^{2} - 3\right) " /> . </p> </p>