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Questions: Algebra BusinessCalculus
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Factor \(\displaystyle - 15 x^{3} + 40 x^{2} - 21 x + 56\).
Factoring out the GCF \(\displaystyle -1\) from each term gives \(\displaystyle -(15 x^{3} - 40 x^{2} + 21 x - 56)\). Grouping the first two terms and factoring out their GCF, \(\displaystyle 5 x^{2}\), gives \(\displaystyle 5 x^{2}(3 x - 8)\). Grouping the last two terms and factoring out their GCF, \(\displaystyle 7\), gives \(\displaystyle 7(3 x - 8)\). The polynomial now has a common binomial factor of \(\displaystyle 3 x - 8\). This gives \(\displaystyle -1[5 x^{2} \left(3 x - 8\right) +7 \cdot \left(3 x - 8\right)] = -\left(3 x - 8\right) \left(5 x^{2} + 7\right)\).
\begin{question}Factor $- 15 x^{3} + 40 x^{2} - 21 x + 56$.
\soln{9cm}{Factoring out the GCF $-1$ from each term gives $-(15 x^{3} - 40 x^{2} + 21 x - 56)$. Grouping the first two terms and factoring out their GCF, $5 x^{2}$, gives $5 x^{2}(3 x - 8)$. Grouping the last two terms and factoring out their GCF, $7$, gives $7(3 x - 8)$. The polynomial now has a common binomial factor of $3 x - 8$. This gives $-1[5 x^{2} \left(3 x - 8\right) +7 \cdot \left(3 x - 8\right)] = -\left(3 x - 8\right) \left(5 x^{2} + 7\right)$. }
\end{question}
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\begin{document}\begin{question}(10pts) The question goes here!
\soln{9cm}{The solution goes here.}
\end{question}\end{document}<p> <p>Factor <img class="equation_image" title=" \displaystyle - 15 x^{3} + 40 x^{2} - 21 x + 56 " src="/equation_images/%20%5Cdisplaystyle%20-%2015%20x%5E%7B3%7D%20%2B%2040%20x%5E%7B2%7D%20-%2021%20x%20%2B%2056%20" alt="LaTeX: \displaystyle - 15 x^{3} + 40 x^{2} - 21 x + 56 " data-equation-content=" \displaystyle - 15 x^{3} + 40 x^{2} - 21 x + 56 " /> . </p> </p><p> <p>Factoring out the 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 " /> from each term gives <img class="equation_image" title=" \displaystyle -(15 x^{3} - 40 x^{2} + 21 x - 56) " src="/equation_images/%20%5Cdisplaystyle%20-%2815%20x%5E%7B3%7D%20-%2040%20x%5E%7B2%7D%20%2B%2021%20x%20-%2056%29%20" alt="LaTeX: \displaystyle -(15 x^{3} - 40 x^{2} + 21 x - 56) " data-equation-content=" \displaystyle -(15 x^{3} - 40 x^{2} + 21 x - 56) " /> . Grouping the first two terms and factoring out their GCF, <img class="equation_image" title=" \displaystyle 5 x^{2} " src="/equation_images/%20%5Cdisplaystyle%205%20x%5E%7B2%7D%20" alt="LaTeX: \displaystyle 5 x^{2} " data-equation-content=" \displaystyle 5 x^{2} " /> , gives <img class="equation_image" title=" \displaystyle 5 x^{2}(3 x - 8) " src="/equation_images/%20%5Cdisplaystyle%205%20x%5E%7B2%7D%283%20x%20-%208%29%20" alt="LaTeX: \displaystyle 5 x^{2}(3 x - 8) " data-equation-content=" \displaystyle 5 x^{2}(3 x - 8) " /> . Grouping the last two terms and factoring out their GCF, <img class="equation_image" title=" \displaystyle 7 " src="/equation_images/%20%5Cdisplaystyle%207%20" alt="LaTeX: \displaystyle 7 " data-equation-content=" \displaystyle 7 " /> , gives <img class="equation_image" title=" \displaystyle 7(3 x - 8) " src="/equation_images/%20%5Cdisplaystyle%207%283%20x%20-%208%29%20" alt="LaTeX: \displaystyle 7(3 x - 8) " data-equation-content=" \displaystyle 7(3 x - 8) " /> . The polynomial now has a common binomial factor of <img class="equation_image" title=" \displaystyle 3 x - 8 " src="/equation_images/%20%5Cdisplaystyle%203%20x%20-%208%20" alt="LaTeX: \displaystyle 3 x - 8 " data-equation-content=" \displaystyle 3 x - 8 " /> . This gives <img class="equation_image" title=" \displaystyle -1[5 x^{2} \left(3 x - 8\right) +7 \cdot \left(3 x - 8\right)] = -\left(3 x - 8\right) \left(5 x^{2} + 7\right) " src="/equation_images/%20%5Cdisplaystyle%20-1%5B5%20x%5E%7B2%7D%20%5Cleft%283%20x%20-%208%5Cright%29%20%2B7%20%5Ccdot%20%5Cleft%283%20x%20-%208%5Cright%29%5D%20%3D%20-%5Cleft%283%20x%20-%208%5Cright%29%20%5Cleft%285%20x%5E%7B2%7D%20%2B%207%5Cright%29%20" alt="LaTeX: \displaystyle -1[5 x^{2} \left(3 x - 8\right) +7 \cdot \left(3 x - 8\right)] = -\left(3 x - 8\right) \left(5 x^{2} + 7\right) " data-equation-content=" \displaystyle -1[5 x^{2} \left(3 x - 8\right) +7 \cdot \left(3 x - 8\right)] = -\left(3 x - 8\right) \left(5 x^{2} + 7\right) " /> . </p> </p>