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Questions: Algebra BusinessCalculus
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Factor \(\displaystyle - 42 x^{3} - 56 x^{2} - 6 x - 8\).
Factoring out the GCF \(\displaystyle -2\) from each term gives \(\displaystyle -2(21 x^{3} + 28 x^{2} + 3 x + 4)\). Grouping the first two terms and factoring out their GCF, \(\displaystyle 7 x^{2}\), gives \(\displaystyle 7 x^{2}(3 x + 4)\). Grouping the last two terms and factoring out their GCF, \(\displaystyle 1\), gives \(\displaystyle 1(3 x + 4)\). The polynomial now has a common binomial factor of \(\displaystyle 3 x + 4\). This gives \(\displaystyle -2[7 x^{2} \left(3 x + 4\right) +1 \cdot \left(3 x + 4\right)] = -2\left(3 x + 4\right) \left(7 x^{2} + 1\right)\).
\begin{question}Factor $- 42 x^{3} - 56 x^{2} - 6 x - 8$.
\soln{9cm}{Factoring out the GCF $-2$ from each term gives $-2(21 x^{3} + 28 x^{2} + 3 x + 4)$. Grouping the first two terms and factoring out their GCF, $7 x^{2}$, gives $7 x^{2}(3 x + 4)$. Grouping the last two terms and factoring out their GCF, $1$, gives $1(3 x + 4)$. The polynomial now has a common binomial factor of $3 x + 4$. This gives $-2[7 x^{2} \left(3 x + 4\right) +1 \cdot \left(3 x + 4\right)] = -2\left(3 x + 4\right) \left(7 x^{2} + 1\right)$. }
\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 - 42 x^{3} - 56 x^{2} - 6 x - 8 " src="/equation_images/%20%5Cdisplaystyle%20-%2042%20x%5E%7B3%7D%20-%2056%20x%5E%7B2%7D%20-%206%20x%20-%208%20" alt="LaTeX: \displaystyle - 42 x^{3} - 56 x^{2} - 6 x - 8 " data-equation-content=" \displaystyle - 42 x^{3} - 56 x^{2} - 6 x - 8 " /> . </p> </p><p> <p>Factoring out the GCF <img class="equation_image" title=" \displaystyle -2 " src="/equation_images/%20%5Cdisplaystyle%20-2%20" alt="LaTeX: \displaystyle -2 " data-equation-content=" \displaystyle -2 " /> from each term gives <img class="equation_image" title=" \displaystyle -2(21 x^{3} + 28 x^{2} + 3 x + 4) " src="/equation_images/%20%5Cdisplaystyle%20-2%2821%20x%5E%7B3%7D%20%2B%2028%20x%5E%7B2%7D%20%2B%203%20x%20%2B%204%29%20" alt="LaTeX: \displaystyle -2(21 x^{3} + 28 x^{2} + 3 x + 4) " data-equation-content=" \displaystyle -2(21 x^{3} + 28 x^{2} + 3 x + 4) " /> . Grouping the first two terms and factoring out their GCF, <img class="equation_image" title=" \displaystyle 7 x^{2} " src="/equation_images/%20%5Cdisplaystyle%207%20x%5E%7B2%7D%20" alt="LaTeX: \displaystyle 7 x^{2} " data-equation-content=" \displaystyle 7 x^{2} " /> , gives <img class="equation_image" title=" \displaystyle 7 x^{2}(3 x + 4) " src="/equation_images/%20%5Cdisplaystyle%207%20x%5E%7B2%7D%283%20x%20%2B%204%29%20" alt="LaTeX: \displaystyle 7 x^{2}(3 x + 4) " data-equation-content=" \displaystyle 7 x^{2}(3 x + 4) " /> . Grouping the last two terms and factoring out their GCF, <img class="equation_image" title=" \displaystyle 1 " src="/equation_images/%20%5Cdisplaystyle%201%20" alt="LaTeX: \displaystyle 1 " data-equation-content=" \displaystyle 1 " /> , gives <img class="equation_image" title=" \displaystyle 1(3 x + 4) " src="/equation_images/%20%5Cdisplaystyle%201%283%20x%20%2B%204%29%20" alt="LaTeX: \displaystyle 1(3 x + 4) " data-equation-content=" \displaystyle 1(3 x + 4) " /> . The polynomial now has a common binomial factor of <img class="equation_image" title=" \displaystyle 3 x + 4 " src="/equation_images/%20%5Cdisplaystyle%203%20x%20%2B%204%20" alt="LaTeX: \displaystyle 3 x + 4 " data-equation-content=" \displaystyle 3 x + 4 " /> . This gives <img class="equation_image" title=" \displaystyle -2[7 x^{2} \left(3 x + 4\right) +1 \cdot \left(3 x + 4\right)] = -2\left(3 x + 4\right) \left(7 x^{2} + 1\right) " src="/equation_images/%20%5Cdisplaystyle%20-2%5B7%20x%5E%7B2%7D%20%5Cleft%283%20x%20%2B%204%5Cright%29%20%2B1%20%5Ccdot%20%5Cleft%283%20x%20%2B%204%5Cright%29%5D%20%3D%20-2%5Cleft%283%20x%20%2B%204%5Cright%29%20%5Cleft%287%20x%5E%7B2%7D%20%2B%201%5Cright%29%20" alt="LaTeX: \displaystyle -2[7 x^{2} \left(3 x + 4\right) +1 \cdot \left(3 x + 4\right)] = -2\left(3 x + 4\right) \left(7 x^{2} + 1\right) " data-equation-content=" \displaystyle -2[7 x^{2} \left(3 x + 4\right) +1 \cdot \left(3 x + 4\right)] = -2\left(3 x + 4\right) \left(7 x^{2} + 1\right) " /> . </p> </p>