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

Please login to create an exam or a quiz.

Algebra
Quadratics
New Random

Factor \(\displaystyle 27 x^{3} + 9 x^{2} - 12 x - 4\).

Grouping the first two terms and factoring out their GCF, \(\displaystyle 9 x^{2}\), gives \(\displaystyle 9 x^{2}(3 x + 1)\). Grouping the last two terms and factoring out their GCF, \(\displaystyle -4\), gives \(\displaystyle -4(3 x + 1)\). The polynomial now has a common binomial factor of \(\displaystyle 3 x + 1\). This gives \(\displaystyle 9 x^{2} \left(3 x + 1\right) -4 \cdot \left(3 x + 1\right) = \left(3 x + 1\right) \left(9 x^{2} - 4\right)\). The quadratic factor can be factored using the difference of squares to give \(\displaystyle \left(3 x - 2\right) \left(3 x + 1\right) \left(3 x + 2\right). \)

Download \(\LaTeX\) 

\begin{question}Factor $27 x^{3} + 9 x^{2} - 12 x - 4$. 
    \soln{9cm}{Grouping the first two terms and factoring out their GCF, $9 x^{2}$, gives $9 x^{2}(3 x + 1)$. Grouping the last two terms and factoring out their GCF, $-4$, gives $-4(3 x + 1)$. The polynomial now has a common binomial factor of $3 x + 1$. This gives $9 x^{2} \left(3 x + 1\right) -4 \cdot \left(3 x + 1\right) = \left(3 x + 1\right) \left(9 x^{2} - 4\right)$. The quadratic factor can be factored using the difference of squares to give $\left(3 x - 2\right) \left(3 x + 1\right) \left(3 x + 2\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 27 x^{3} + 9 x^{2} - 12 x - 4 " src="/equation_images/%20%5Cdisplaystyle%2027%20x%5E%7B3%7D%20%2B%209%20x%5E%7B2%7D%20-%2012%20x%20-%204%20" alt="LaTeX:  \displaystyle 27 x^{3} + 9 x^{2} - 12 x - 4 " data-equation-content=" \displaystyle 27 x^{3} + 9 x^{2} - 12 x - 4 " /> . </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}(3 x + 1) " src="/equation_images/%20%5Cdisplaystyle%209%20x%5E%7B2%7D%283%20x%20%2B%201%29%20" alt="LaTeX:  \displaystyle 9 x^{2}(3 x + 1) " data-equation-content=" \displaystyle 9 x^{2}(3 x + 1) " /> . Grouping the last two terms and factoring out their GCF,  <img class="equation_image" title=" \displaystyle -4 " src="/equation_images/%20%5Cdisplaystyle%20-4%20" alt="LaTeX:  \displaystyle -4 " data-equation-content=" \displaystyle -4 " /> , gives  <img class="equation_image" title=" \displaystyle -4(3 x + 1) " src="/equation_images/%20%5Cdisplaystyle%20-4%283%20x%20%2B%201%29%20" alt="LaTeX:  \displaystyle -4(3 x + 1) " data-equation-content=" \displaystyle -4(3 x + 1) " /> . The polynomial now has a common binomial factor of  <img class="equation_image" title=" \displaystyle 3 x + 1 " src="/equation_images/%20%5Cdisplaystyle%203%20x%20%2B%201%20" alt="LaTeX:  \displaystyle 3 x + 1 " data-equation-content=" \displaystyle 3 x + 1 " /> . This gives  <img class="equation_image" title=" \displaystyle 9 x^{2} \left(3 x + 1\right) -4 \cdot \left(3 x + 1\right) = \left(3 x + 1\right) \left(9 x^{2} - 4\right) " src="/equation_images/%20%5Cdisplaystyle%209%20x%5E%7B2%7D%20%5Cleft%283%20x%20%2B%201%5Cright%29%20-4%20%5Ccdot%20%5Cleft%283%20x%20%2B%201%5Cright%29%20%3D%20%5Cleft%283%20x%20%2B%201%5Cright%29%20%5Cleft%289%20x%5E%7B2%7D%20-%204%5Cright%29%20" alt="LaTeX:  \displaystyle 9 x^{2} \left(3 x + 1\right) -4 \cdot \left(3 x + 1\right) = \left(3 x + 1\right) \left(9 x^{2} - 4\right) " data-equation-content=" \displaystyle 9 x^{2} \left(3 x + 1\right) -4 \cdot \left(3 x + 1\right) = \left(3 x + 1\right) \left(9 x^{2} - 4\right) " /> . The quadratic factor can be factored using the difference of squares to give  <img class="equation_image" title=" \displaystyle \left(3 x - 2\right) \left(3 x + 1\right) \left(3 x + 2\right).  " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%283%20x%20-%202%5Cright%29%20%5Cleft%283%20x%20%2B%201%5Cright%29%20%5Cleft%283%20x%20%2B%202%5Cright%29.%20%20" alt="LaTeX:  \displaystyle \left(3 x - 2\right) \left(3 x + 1\right) \left(3 x + 2\right).  " data-equation-content=" \displaystyle \left(3 x - 2\right) \left(3 x + 1\right) \left(3 x + 2\right).  " /> </p> </p>