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Questions: Algebra BusinessCalculus

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Algebra
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Factor \(\displaystyle 18 x^{3} + 10 x^{2} + 27 x + 15\).

Grouping the first two terms and factoring out their GCF, \(\displaystyle 2 x^{2}\), gives \(\displaystyle 2 x^{2}(9 x + 5)\). Grouping the last two terms and factoring out their GCF, \(\displaystyle 3\), gives \(\displaystyle 3(9 x + 5)\). The polynomial now has a common binomial factor of \(\displaystyle 9 x + 5\). This gives \(\displaystyle 2 x^{2} \left(9 x + 5\right) +3 \cdot \left(9 x + 5\right) = \left(9 x + 5\right) \left(2 x^{2} + 3\right)\).

Download \(\LaTeX\) 

\begin{question}Factor $18 x^{3} + 10 x^{2} + 27 x + 15$. 
    \soln{9cm}{Grouping the first two terms and factoring out their GCF, $2 x^{2}$, gives $2 x^{2}(9 x + 5)$. Grouping the last two terms and factoring out their GCF, $3$, gives $3(9 x + 5)$. The polynomial now has a common binomial factor of $9 x + 5$. This gives $2 x^{2} \left(9 x + 5\right) +3 \cdot \left(9 x + 5\right) = \left(9 x + 5\right) \left(2 x^{2} + 3\right)$. }

\end{question}

Download Question and Solution Environment\(\LaTeX\)
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HTML for Canvas 
<p> <p>Factor  <img class="equation_image" title=" \displaystyle 18 x^{3} + 10 x^{2} + 27 x + 15 " src="/equation_images/%20%5Cdisplaystyle%2018%20x%5E%7B3%7D%20%2B%2010%20x%5E%7B2%7D%20%2B%2027%20x%20%2B%2015%20" alt="LaTeX:  \displaystyle 18 x^{3} + 10 x^{2} + 27 x + 15 " data-equation-content=" \displaystyle 18 x^{3} + 10 x^{2} + 27 x + 15 " /> . </p> </p>
HTML for Canvas
<p> <p>Grouping the first two terms and factoring out their GCF,  <img class="equation_image" title=" \displaystyle 2 x^{2} " src="/equation_images/%20%5Cdisplaystyle%202%20x%5E%7B2%7D%20" alt="LaTeX:  \displaystyle 2 x^{2} " data-equation-content=" \displaystyle 2 x^{2} " /> , gives  <img class="equation_image" title=" \displaystyle 2 x^{2}(9 x + 5) " src="/equation_images/%20%5Cdisplaystyle%202%20x%5E%7B2%7D%289%20x%20%2B%205%29%20" alt="LaTeX:  \displaystyle 2 x^{2}(9 x + 5) " data-equation-content=" \displaystyle 2 x^{2}(9 x + 5) " /> . Grouping the last two terms and factoring out their GCF,  <img class="equation_image" title=" \displaystyle 3 " src="/equation_images/%20%5Cdisplaystyle%203%20" alt="LaTeX:  \displaystyle 3 " data-equation-content=" \displaystyle 3 " /> , gives  <img class="equation_image" title=" \displaystyle 3(9 x + 5) " src="/equation_images/%20%5Cdisplaystyle%203%289%20x%20%2B%205%29%20" alt="LaTeX:  \displaystyle 3(9 x + 5) " data-equation-content=" \displaystyle 3(9 x + 5) " /> . The polynomial now has a common binomial factor of  <img class="equation_image" title=" \displaystyle 9 x + 5 " src="/equation_images/%20%5Cdisplaystyle%209%20x%20%2B%205%20" alt="LaTeX:  \displaystyle 9 x + 5 " data-equation-content=" \displaystyle 9 x + 5 " /> . This gives  <img class="equation_image" title=" \displaystyle 2 x^{2} \left(9 x + 5\right) +3 \cdot \left(9 x + 5\right) = \left(9 x + 5\right) \left(2 x^{2} + 3\right) " src="/equation_images/%20%5Cdisplaystyle%202%20x%5E%7B2%7D%20%5Cleft%289%20x%20%2B%205%5Cright%29%20%2B3%20%5Ccdot%20%5Cleft%289%20x%20%2B%205%5Cright%29%20%3D%20%5Cleft%289%20x%20%2B%205%5Cright%29%20%5Cleft%282%20x%5E%7B2%7D%20%2B%203%5Cright%29%20" alt="LaTeX:  \displaystyle 2 x^{2} \left(9 x + 5\right) +3 \cdot \left(9 x + 5\right) = \left(9 x + 5\right) \left(2 x^{2} + 3\right) " data-equation-content=" \displaystyle 2 x^{2} \left(9 x + 5\right) +3 \cdot \left(9 x + 5\right) = \left(9 x + 5\right) \left(2 x^{2} + 3\right) " /> . </p> </p>