\(\text{www.the}\beta\text{etafunction.com}\)
Home
Login
Questions: Algebra BusinessCalculus

Please login to create an exam or a quiz.

Algebra
Quadratics
New Random

Factor \(\displaystyle 45 x^{3} + 54 x^{2} + 20 x + 24\).

Grouping the first two terms and factoring out their GCF, \(\displaystyle 9 x^{2}\), gives \(\displaystyle 9 x^{2}(5 x + 6)\). Grouping the last two terms and factoring out their GCF, \(\displaystyle 4\), gives \(\displaystyle 4(5 x + 6)\). The polynomial now has a common binomial factor of \(\displaystyle 5 x + 6\). This gives \(\displaystyle 9 x^{2} \left(5 x + 6\right) +4 \cdot \left(5 x + 6\right) = \left(5 x + 6\right) \left(9 x^{2} + 4\right)\).

Download \(\LaTeX\) 

\begin{question}Factor $45 x^{3} + 54 x^{2} + 20 x + 24$. 
    \soln{9cm}{Grouping the first two terms and factoring out their GCF, $9 x^{2}$, gives $9 x^{2}(5 x + 6)$. Grouping the last two terms and factoring out their GCF, $4$, gives $4(5 x + 6)$. The polynomial now has a common binomial factor of $5 x + 6$. This gives $9 x^{2} \left(5 x + 6\right) +4 \cdot \left(5 x + 6\right) = \left(5 x + 6\right) \left(9 x^{2} + 4\right)$. }

\end{question}

Download Question and Solution Environment\(\LaTeX\)
\documentclass{article}
\usepackage{tikz}
\usepackage{amsmath}
\usepackage[margin=2cm]{geometry}
\usepackage{tcolorbox}

\newcounter{ExamNumber}
\newcounter{questioncount}
\stepcounter{questioncount}

\newenvironment{question}{{\noindent\bfseries Question \arabic{questioncount}.}}{\stepcounter{questioncount}}
\renewcommand{\labelenumi}{{\bfseries (\alph{enumi})}}

\newif\ifShowSolution
\newcommand{\soln}[2]{%
\ifShowSolution%
\noindent\begin{tcolorbox}[colframe=blue,title=Solution]#2\end{tcolorbox}\else%
\vspace{#1}%
\fi%
}%
\newcommand{\hideifShowSolution}[1]{%
\ifShowSolution%
%
\else%
#1%
\fi%
}%
\everymath{\displaystyle}
\ShowSolutiontrue

\begin{document}\begin{question}(10pts) The question goes here!
    \soln{9cm}{The solution goes here.}

\end{question}\end{document}
HTML for Canvas 
<p> <p>Factor  <img class="equation_image" title=" \displaystyle 45 x^{3} + 54 x^{2} + 20 x + 24 " src="/equation_images/%20%5Cdisplaystyle%2045%20x%5E%7B3%7D%20%2B%2054%20x%5E%7B2%7D%20%2B%2020%20x%20%2B%2024%20" alt="LaTeX:  \displaystyle 45 x^{3} + 54 x^{2} + 20 x + 24 " data-equation-content=" \displaystyle 45 x^{3} + 54 x^{2} + 20 x + 24 " /> . </p> </p>
HTML for Canvas
<p> <p>Grouping the first two terms and factoring out their GCF,  <img class="equation_image" title=" \displaystyle 9 x^{2} " src="/equation_images/%20%5Cdisplaystyle%209%20x%5E%7B2%7D%20" alt="LaTeX:  \displaystyle 9 x^{2} " data-equation-content=" \displaystyle 9 x^{2} " /> , gives  <img class="equation_image" title=" \displaystyle 9 x^{2}(5 x + 6) " src="/equation_images/%20%5Cdisplaystyle%209%20x%5E%7B2%7D%285%20x%20%2B%206%29%20" alt="LaTeX:  \displaystyle 9 x^{2}(5 x + 6) " data-equation-content=" \displaystyle 9 x^{2}(5 x + 6) " /> . Grouping the last two terms and factoring out their GCF,  <img class="equation_image" title=" \displaystyle 4 " src="/equation_images/%20%5Cdisplaystyle%204%20" alt="LaTeX:  \displaystyle 4 " data-equation-content=" \displaystyle 4 " /> , gives  <img class="equation_image" title=" \displaystyle 4(5 x + 6) " src="/equation_images/%20%5Cdisplaystyle%204%285%20x%20%2B%206%29%20" alt="LaTeX:  \displaystyle 4(5 x + 6) " data-equation-content=" \displaystyle 4(5 x + 6) " /> . The polynomial now has a common binomial factor of  <img class="equation_image" title=" \displaystyle 5 x + 6 " src="/equation_images/%20%5Cdisplaystyle%205%20x%20%2B%206%20" alt="LaTeX:  \displaystyle 5 x + 6 " data-equation-content=" \displaystyle 5 x + 6 " /> . This gives  <img class="equation_image" title=" \displaystyle 9 x^{2} \left(5 x + 6\right) +4 \cdot \left(5 x + 6\right) = \left(5 x + 6\right) \left(9 x^{2} + 4\right) " src="/equation_images/%20%5Cdisplaystyle%209%20x%5E%7B2%7D%20%5Cleft%285%20x%20%2B%206%5Cright%29%20%2B4%20%5Ccdot%20%5Cleft%285%20x%20%2B%206%5Cright%29%20%3D%20%5Cleft%285%20x%20%2B%206%5Cright%29%20%5Cleft%289%20x%5E%7B2%7D%20%2B%204%5Cright%29%20" alt="LaTeX:  \displaystyle 9 x^{2} \left(5 x + 6\right) +4 \cdot \left(5 x + 6\right) = \left(5 x + 6\right) \left(9 x^{2} + 4\right) " data-equation-content=" \displaystyle 9 x^{2} \left(5 x + 6\right) +4 \cdot \left(5 x + 6\right) = \left(5 x + 6\right) \left(9 x^{2} + 4\right) " /> . </p> </p>