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Questions: Algebra BusinessCalculus

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Algebra
Quadratics
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Factor \(\displaystyle 15 x^{3} - 35 x^{2} + 24 x - 56\).

Grouping the first two terms and factoring out their GCF, \(\displaystyle 5 x^{2}\), gives \(\displaystyle 5 x^{2}(3 x - 7)\). Grouping the last two terms and factoring out their GCF, \(\displaystyle 8\), gives \(\displaystyle 8(3 x - 7)\). The polynomial now has a common binomial factor of \(\displaystyle 3 x - 7\). This gives \(\displaystyle 5 x^{2} \left(3 x - 7\right) +8 \cdot \left(3 x - 7\right) = \left(3 x - 7\right) \left(5 x^{2} + 8\right)\).

Download \(\LaTeX\) 

\begin{question}Factor $15 x^{3} - 35 x^{2} + 24 x - 56$. 
    \soln{9cm}{Grouping the first two terms and factoring out their GCF, $5 x^{2}$, gives $5 x^{2}(3 x - 7)$. Grouping the last two terms and factoring out their GCF, $8$, gives $8(3 x - 7)$. The polynomial now has a common binomial factor of $3 x - 7$. This gives $5 x^{2} \left(3 x - 7\right) +8 \cdot \left(3 x - 7\right) = \left(3 x - 7\right) \left(5 x^{2} + 8\right)$. }

\end{question}

Download Question and Solution Environment\(\LaTeX\)
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HTML for Canvas 
<p> <p>Factor  <img class="equation_image" title=" \displaystyle 15 x^{3} - 35 x^{2} + 24 x - 56 " src="/equation_images/%20%5Cdisplaystyle%2015%20x%5E%7B3%7D%20-%2035%20x%5E%7B2%7D%20%2B%2024%20x%20-%2056%20" alt="LaTeX:  \displaystyle 15 x^{3} - 35 x^{2} + 24 x - 56 " data-equation-content=" \displaystyle 15 x^{3} - 35 x^{2} + 24 x - 56 " /> . </p> </p>
HTML for Canvas
<p> <p>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 - 7) " src="/equation_images/%20%5Cdisplaystyle%205%20x%5E%7B2%7D%283%20x%20-%207%29%20" alt="LaTeX:  \displaystyle 5 x^{2}(3 x - 7) " data-equation-content=" \displaystyle 5 x^{2}(3 x - 7) " /> . Grouping the last two terms and factoring out their GCF,  <img class="equation_image" title=" \displaystyle 8 " src="/equation_images/%20%5Cdisplaystyle%208%20" alt="LaTeX:  \displaystyle 8 " data-equation-content=" \displaystyle 8 " /> , gives  <img class="equation_image" title=" \displaystyle 8(3 x - 7) " src="/equation_images/%20%5Cdisplaystyle%208%283%20x%20-%207%29%20" alt="LaTeX:  \displaystyle 8(3 x - 7) " data-equation-content=" \displaystyle 8(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-%207%20" alt="LaTeX:  \displaystyle 3 x - 7 " data-equation-content=" \displaystyle 3 x - 7 " /> . This gives  <img class="equation_image" title=" \displaystyle 5 x^{2} \left(3 x - 7\right) +8 \cdot \left(3 x - 7\right) = \left(3 x - 7\right) \left(5 x^{2} + 8\right) " src="/equation_images/%20%5Cdisplaystyle%205%20x%5E%7B2%7D%20%5Cleft%283%20x%20-%207%5Cright%29%20%2B8%20%5Ccdot%20%5Cleft%283%20x%20-%207%5Cright%29%20%3D%20%5Cleft%283%20x%20-%207%5Cright%29%20%5Cleft%285%20x%5E%7B2%7D%20%2B%208%5Cright%29%20" alt="LaTeX:  \displaystyle 5 x^{2} \left(3 x - 7\right) +8 \cdot \left(3 x - 7\right) = \left(3 x - 7\right) \left(5 x^{2} + 8\right) " data-equation-content=" \displaystyle 5 x^{2} \left(3 x - 7\right) +8 \cdot \left(3 x - 7\right) = \left(3 x - 7\right) \left(5 x^{2} + 8\right) " /> . </p> </p>