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Factor \(\displaystyle 35 x^{3} - 15 x^{2} + 70 x - 30\).

Factoring out the GCF \(\displaystyle 5\) from each term gives \(\displaystyle 5(7 x^{3} - 3 x^{2} + 14 x - 6)\). Grouping the first two terms and factoring out their GCF, \(\displaystyle x^{2}\), gives \(\displaystyle x^{2}(7 x - 3)\). Grouping the last two terms and factoring out their GCF, \(\displaystyle 2\), gives \(\displaystyle 2(7 x - 3)\). The polynomial now has a common binomial factor of \(\displaystyle 7 x - 3\). This gives \(\displaystyle 5[x^{2} \left(7 x - 3\right) +2 \cdot \left(7 x - 3\right)] = 5\left(7 x - 3\right) \left(x^{2} + 2\right)\).

Download \(\LaTeX\) 

\begin{question}Factor $35 x^{3} - 15 x^{2} + 70 x - 30$. 
    \soln{9cm}{Factoring out the GCF $5$ from each term gives $5(7 x^{3} - 3 x^{2} + 14 x - 6)$. Grouping the first two terms and factoring out their GCF, $x^{2}$, gives $x^{2}(7 x - 3)$. Grouping the last two terms and factoring out their GCF, $2$, gives $2(7 x - 3)$. The polynomial now has a common binomial factor of $7 x - 3$. This gives $5[x^{2} \left(7 x - 3\right) +2 \cdot \left(7 x - 3\right)] = 5\left(7 x - 3\right) \left(x^{2} + 2\right)$. }

\end{question}

Download Question and Solution Environment\(\LaTeX\)
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HTML for Canvas 
<p> <p>Factor  <img class="equation_image" title=" \displaystyle 35 x^{3} - 15 x^{2} + 70 x - 30 " src="/equation_images/%20%5Cdisplaystyle%2035%20x%5E%7B3%7D%20-%2015%20x%5E%7B2%7D%20%2B%2070%20x%20-%2030%20" alt="LaTeX:  \displaystyle 35 x^{3} - 15 x^{2} + 70 x - 30 " data-equation-content=" \displaystyle 35 x^{3} - 15 x^{2} + 70 x - 30 " /> . </p> </p>
HTML for Canvas
<p> <p>Factoring out the GCF  <img class="equation_image" title=" \displaystyle 5 " src="/equation_images/%20%5Cdisplaystyle%205%20" alt="LaTeX:  \displaystyle 5 " data-equation-content=" \displaystyle 5 " />  from each term gives  <img class="equation_image" title=" \displaystyle 5(7 x^{3} - 3 x^{2} + 14 x - 6) " src="/equation_images/%20%5Cdisplaystyle%205%287%20x%5E%7B3%7D%20-%203%20x%5E%7B2%7D%20%2B%2014%20x%20-%206%29%20" alt="LaTeX:  \displaystyle 5(7 x^{3} - 3 x^{2} + 14 x - 6) " data-equation-content=" \displaystyle 5(7 x^{3} - 3 x^{2} + 14 x - 6) " /> . 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}(7 x - 3) " src="/equation_images/%20%5Cdisplaystyle%20x%5E%7B2%7D%287%20x%20-%203%29%20" alt="LaTeX:  \displaystyle x^{2}(7 x - 3) " data-equation-content=" \displaystyle x^{2}(7 x - 3) " /> . Grouping the last two terms and factoring out their GCF,  <img class="equation_image" title=" \displaystyle 2 " src="/equation_images/%20%5Cdisplaystyle%202%20" alt="LaTeX:  \displaystyle 2 " data-equation-content=" \displaystyle 2 " /> , gives  <img class="equation_image" title=" \displaystyle 2(7 x - 3) " src="/equation_images/%20%5Cdisplaystyle%202%287%20x%20-%203%29%20" alt="LaTeX:  \displaystyle 2(7 x - 3) " data-equation-content=" \displaystyle 2(7 x - 3) " /> . The polynomial now has a common binomial factor of  <img class="equation_image" title=" \displaystyle 7 x - 3 " src="/equation_images/%20%5Cdisplaystyle%207%20x%20-%203%20" alt="LaTeX:  \displaystyle 7 x - 3 " data-equation-content=" \displaystyle 7 x - 3 " /> . This gives  <img class="equation_image" title=" \displaystyle 5[x^{2} \left(7 x - 3\right) +2 \cdot \left(7 x - 3\right)] = 5\left(7 x - 3\right) \left(x^{2} + 2\right) " src="/equation_images/%20%5Cdisplaystyle%205%5Bx%5E%7B2%7D%20%5Cleft%287%20x%20-%203%5Cright%29%20%2B2%20%5Ccdot%20%5Cleft%287%20x%20-%203%5Cright%29%5D%20%3D%205%5Cleft%287%20x%20-%203%5Cright%29%20%5Cleft%28x%5E%7B2%7D%20%2B%202%5Cright%29%20" alt="LaTeX:  \displaystyle 5[x^{2} \left(7 x - 3\right) +2 \cdot \left(7 x - 3\right)] = 5\left(7 x - 3\right) \left(x^{2} + 2\right) " data-equation-content=" \displaystyle 5[x^{2} \left(7 x - 3\right) +2 \cdot \left(7 x - 3\right)] = 5\left(7 x - 3\right) \left(x^{2} + 2\right) " /> . </p> </p>