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Factoring out the GCF \(\displaystyle 10\) from each term gives \(\displaystyle 10(3 x^{3} + 7 x^{2} - 3 x - 7)\). Grouping the first two terms and factoring out their GCF, \(\displaystyle x^{2}\), gives \(\displaystyle x^{2}(3 x + 7)\). Grouping the last two terms and factoring out their GCF, \(\displaystyle -1\), gives \(\displaystyle -1(3 x + 7)\). The polynomial now has a common binomial factor of \(\displaystyle 3 x + 7\). This gives \(\displaystyle 10[x^{2} \left(3 x + 7\right) -1 \cdot \left(3 x + 7\right)] = 10\left(3 x + 7\right) \left(x^{2} - 1\right)\). The quadratic factor can be factored using the difference of squares to give \(\displaystyle 10\left(x - 1\right) \left(x + 1\right) \left(3 x + 7\right). \)

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\begin{question}Factor $30 x^{3} + 70 x^{2} - 30 x - 70$. 
    \soln{9cm}{Factoring out the GCF $10$ from each term gives $10(3 x^{3} + 7 x^{2} - 3 x - 7)$. Grouping the first two terms and factoring out their GCF, $x^{2}$, gives $x^{2}(3 x + 7)$. Grouping the last two terms and factoring out their GCF, $-1$, gives $-1(3 x + 7)$. The polynomial now has a common binomial factor of $3 x + 7$. This gives $10[x^{2} \left(3 x + 7\right) -1 \cdot \left(3 x + 7\right)] = 10\left(3 x + 7\right) \left(x^{2} - 1\right)$. The quadratic factor can be factored using the difference of squares to give $10\left(x - 1\right) \left(x + 1\right) \left(3 x + 7\right). $}

\end{question}

Download Question and Solution Environment\(\LaTeX\)

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\begin{document}\begin{question}(10pts) The question goes here!
    \soln{9cm}{The solution goes here.}

\end{question}\end{document}

HTML for Canvas

<p> <p>Factor  <img class="equation_image" title=" \displaystyle 30 x^{3} + 70 x^{2} - 30 x - 70 " src="/equation_images/%20%5Cdisplaystyle%2030%20x%5E%7B3%7D%20%2B%2070%20x%5E%7B2%7D%20-%2030%20x%20-%2070%20" alt="LaTeX:  \displaystyle 30 x^{3} + 70 x^{2} - 30 x - 70 " data-equation-content=" \displaystyle 30 x^{3} + 70 x^{2} - 30 x - 70 " /> . </p> </p>

HTML for Canvas

<p> <p>Factoring out the GCF  <img class="equation_image" title=" \displaystyle 10 " src="/equation_images/%20%5Cdisplaystyle%2010%20" alt="LaTeX:  \displaystyle 10 " data-equation-content=" \displaystyle 10 " />  from each term gives  <img class="equation_image" title=" \displaystyle 10(3 x^{3} + 7 x^{2} - 3 x - 7) " src="/equation_images/%20%5Cdisplaystyle%2010%283%20x%5E%7B3%7D%20%2B%207%20x%5E%7B2%7D%20-%203%20x%20-%207%29%20" alt="LaTeX:  \displaystyle 10(3 x^{3} + 7 x^{2} - 3 x - 7) " data-equation-content=" \displaystyle 10(3 x^{3} + 7 x^{2} - 3 x - 7) " /> . 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}(3 x + 7) " src="/equation_images/%20%5Cdisplaystyle%20x%5E%7B2%7D%283%20x%20%2B%207%29%20" alt="LaTeX:  \displaystyle x^{2}(3 x + 7) " data-equation-content=" \displaystyle x^{2}(3 x + 7) " /> . Grouping the last two terms and factoring out their GCF,  <img class="equation_image" title=" \displaystyle -1 " src="/equation_images/%20%5Cdisplaystyle%20-1%20" alt="LaTeX:  \displaystyle -1 " data-equation-content=" \displaystyle -1 " /> , gives  <img class="equation_image" title=" \displaystyle -1(3 x + 7) " src="/equation_images/%20%5Cdisplaystyle%20-1%283%20x%20%2B%207%29%20" alt="LaTeX:  \displaystyle -1(3 x + 7) " data-equation-content=" \displaystyle -1(3 x + 7) " /> . The polynomial now has a common binomial factor of  <img class="equation_image" title=" \displaystyle 3 x + 7 " src="/equation_images/%20%5Cdisplaystyle%203%20x%20%2B%207%20" alt="LaTeX:  \displaystyle 3 x + 7 " data-equation-content=" \displaystyle 3 x + 7 " /> . This gives  <img class="equation_image" title=" \displaystyle 10[x^{2} \left(3 x + 7\right) -1 \cdot \left(3 x + 7\right)] = 10\left(3 x + 7\right) \left(x^{2} - 1\right) " src="/equation_images/%20%5Cdisplaystyle%2010%5Bx%5E%7B2%7D%20%5Cleft%283%20x%20%2B%207%5Cright%29%20-1%20%5Ccdot%20%5Cleft%283%20x%20%2B%207%5Cright%29%5D%20%3D%2010%5Cleft%283%20x%20%2B%207%5Cright%29%20%5Cleft%28x%5E%7B2%7D%20-%201%5Cright%29%20" alt="LaTeX:  \displaystyle 10[x^{2} \left(3 x + 7\right) -1 \cdot \left(3 x + 7\right)] = 10\left(3 x + 7\right) \left(x^{2} - 1\right) " data-equation-content=" \displaystyle 10[x^{2} \left(3 x + 7\right) -1 \cdot \left(3 x + 7\right)] = 10\left(3 x + 7\right) \left(x^{2} - 1\right) " /> . The quadratic factor can be factored using the difference of squares to give  <img class="equation_image" title=" \displaystyle 10\left(x - 1\right) \left(x + 1\right) \left(3 x + 7\right).  " src="/equation_images/%20%5Cdisplaystyle%2010%5Cleft%28x%20-%201%5Cright%29%20%5Cleft%28x%20%2B%201%5Cright%29%20%5Cleft%283%20x%20%2B%207%5Cright%29.%20%20" alt="LaTeX:  \displaystyle 10\left(x - 1\right) \left(x + 1\right) \left(3 x + 7\right).  " data-equation-content=" \displaystyle 10\left(x - 1\right) \left(x + 1\right) \left(3 x + 7\right).  " /> </p> </p>