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

Factoring out the GCF \(\displaystyle -35\) from each term gives \(\displaystyle -35(x^{3} - x^{2} - x + 1)\). Grouping the first two terms and factoring out their GCF, \(\displaystyle x^{2}\), gives \(\displaystyle x^{2}(x - 1)\). Grouping the last two terms and factoring out their GCF, \(\displaystyle -1\), gives \(\displaystyle -1(x - 1)\). The polynomial now has a common binomial factor of \(\displaystyle x - 1\). This gives \(\displaystyle -35[x^{2} \left(x - 1\right) -1 \cdot \left(x - 1\right)] = -35\left(x - 1\right) \left(x^{2} - 1\right)\). The quadratic factor can be factored using the difference of squares to give \(\displaystyle -35\left(x - 1\right)^{2} \left(x + 1\right). \)

Download \(\LaTeX\) 

\begin{question}Factor $- 35 x^{3} + 35 x^{2} + 35 x - 35$. 
    \soln{9cm}{Factoring out the GCF $-35$ from each term gives $-35(x^{3} - x^{2} - x + 1)$. Grouping the first two terms and factoring out their GCF, $x^{2}$, gives $x^{2}(x - 1)$. Grouping the last two terms and factoring out their GCF, $-1$, gives $-1(x - 1)$. The polynomial now has a common binomial factor of $x - 1$. This gives $-35[x^{2} \left(x - 1\right) -1 \cdot \left(x - 1\right)] = -35\left(x - 1\right) \left(x^{2} - 1\right)$. The quadratic factor can be factored using the difference of squares to give $-35\left(x - 1\right)^{2} \left(x + 1\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} + 35 x^{2} + 35 x - 35 " src="/equation_images/%20%5Cdisplaystyle%20-%2035%20x%5E%7B3%7D%20%2B%2035%20x%5E%7B2%7D%20%2B%2035%20x%20-%2035%20" alt="LaTeX:  \displaystyle - 35 x^{3} + 35 x^{2} + 35 x - 35 " data-equation-content=" \displaystyle - 35 x^{3} + 35 x^{2} + 35 x - 35 " /> . </p> </p>
HTML for Canvas
<p> <p>Factoring out the GCF  <img class="equation_image" title=" \displaystyle -35 " src="/equation_images/%20%5Cdisplaystyle%20-35%20" alt="LaTeX:  \displaystyle -35 " data-equation-content=" \displaystyle -35 " />  from each term gives  <img class="equation_image" title=" \displaystyle -35(x^{3} - x^{2} - x + 1) " src="/equation_images/%20%5Cdisplaystyle%20-35%28x%5E%7B3%7D%20-%20x%5E%7B2%7D%20-%20x%20%2B%201%29%20" alt="LaTeX:  \displaystyle -35(x^{3} - x^{2} - x + 1) " data-equation-content=" \displaystyle -35(x^{3} - x^{2} - x + 1) " /> . 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 - 1) " src="/equation_images/%20%5Cdisplaystyle%20x%5E%7B2%7D%28x%20-%201%29%20" alt="LaTeX:  \displaystyle x^{2}(x - 1) " data-equation-content=" \displaystyle x^{2}(x - 1) " /> . 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(x - 1) " src="/equation_images/%20%5Cdisplaystyle%20-1%28x%20-%201%29%20" alt="LaTeX:  \displaystyle -1(x - 1) " data-equation-content=" \displaystyle -1(x - 1) " /> . The polynomial now has a common binomial factor of  <img class="equation_image" title=" \displaystyle x - 1 " src="/equation_images/%20%5Cdisplaystyle%20x%20-%201%20" alt="LaTeX:  \displaystyle x - 1 " data-equation-content=" \displaystyle x - 1 " /> . This gives  <img class="equation_image" title=" \displaystyle -35[x^{2} \left(x - 1\right) -1 \cdot \left(x - 1\right)] = -35\left(x - 1\right) \left(x^{2} - 1\right) " src="/equation_images/%20%5Cdisplaystyle%20-35%5Bx%5E%7B2%7D%20%5Cleft%28x%20-%201%5Cright%29%20-1%20%5Ccdot%20%5Cleft%28x%20-%201%5Cright%29%5D%20%3D%20-35%5Cleft%28x%20-%201%5Cright%29%20%5Cleft%28x%5E%7B2%7D%20-%201%5Cright%29%20" alt="LaTeX:  \displaystyle -35[x^{2} \left(x - 1\right) -1 \cdot \left(x - 1\right)] = -35\left(x - 1\right) \left(x^{2} - 1\right) " data-equation-content=" \displaystyle -35[x^{2} \left(x - 1\right) -1 \cdot \left(x - 1\right)] = -35\left(x - 1\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 -35\left(x - 1\right)^{2} \left(x + 1\right).  " src="/equation_images/%20%5Cdisplaystyle%20-35%5Cleft%28x%20-%201%5Cright%29%5E%7B2%7D%20%5Cleft%28x%20%2B%201%5Cright%29.%20%20" alt="LaTeX:  \displaystyle -35\left(x - 1\right)^{2} \left(x + 1\right).  " data-equation-content=" \displaystyle -35\left(x - 1\right)^{2} \left(x + 1\right).  " /> </p> </p>