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

Please login to create an exam or a quiz.

Algebra
Quadratics
New Random

Factor \(\displaystyle 24 x^{3} - 32 x^{2} - 30 x + 40\).

Factoring out the GCF \(\displaystyle 2\) from each term gives \(\displaystyle 2(12 x^{3} - 16 x^{2} - 15 x + 20)\). Grouping the first two terms and factoring out their GCF, \(\displaystyle 4 x^{2}\), gives \(\displaystyle 4 x^{2}(3 x - 4)\). Grouping the last two terms and factoring out their GCF, \(\displaystyle -5\), gives \(\displaystyle -5(3 x - 4)\). The polynomial now has a common binomial factor of \(\displaystyle 3 x - 4\). This gives \(\displaystyle 2[4 x^{2} \left(3 x - 4\right) -5 \cdot \left(3 x - 4\right)] = 2\left(3 x - 4\right) \left(4 x^{2} - 5\right)\).

Download \(\LaTeX\) 

\begin{question}Factor $24 x^{3} - 32 x^{2} - 30 x + 40$. 
    \soln{9cm}{Factoring out the GCF $2$ from each term gives $2(12 x^{3} - 16 x^{2} - 15 x + 20)$. Grouping the first two terms and factoring out their GCF, $4 x^{2}$, gives $4 x^{2}(3 x - 4)$. Grouping the last two terms and factoring out their GCF, $-5$, gives $-5(3 x - 4)$. The polynomial now has a common binomial factor of $3 x - 4$. This gives $2[4 x^{2} \left(3 x - 4\right) -5 \cdot \left(3 x - 4\right)] = 2\left(3 x - 4\right) \left(4 x^{2} - 5\right)$. }

\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 24 x^{3} - 32 x^{2} - 30 x + 40 " src="/equation_images/%20%5Cdisplaystyle%2024%20x%5E%7B3%7D%20-%2032%20x%5E%7B2%7D%20-%2030%20x%20%2B%2040%20" alt="LaTeX:  \displaystyle 24 x^{3} - 32 x^{2} - 30 x + 40 " data-equation-content=" \displaystyle 24 x^{3} - 32 x^{2} - 30 x + 40 " /> . </p> </p>
HTML for Canvas
<p> <p>Factoring out the GCF  <img class="equation_image" title=" \displaystyle 2 " src="/equation_images/%20%5Cdisplaystyle%202%20" alt="LaTeX:  \displaystyle 2 " data-equation-content=" \displaystyle 2 " />  from each term gives  <img class="equation_image" title=" \displaystyle 2(12 x^{3} - 16 x^{2} - 15 x + 20) " src="/equation_images/%20%5Cdisplaystyle%202%2812%20x%5E%7B3%7D%20-%2016%20x%5E%7B2%7D%20-%2015%20x%20%2B%2020%29%20" alt="LaTeX:  \displaystyle 2(12 x^{3} - 16 x^{2} - 15 x + 20) " data-equation-content=" \displaystyle 2(12 x^{3} - 16 x^{2} - 15 x + 20) " /> . Grouping the first two terms and factoring out their GCF,  <img class="equation_image" title=" \displaystyle 4 x^{2} " src="/equation_images/%20%5Cdisplaystyle%204%20x%5E%7B2%7D%20" alt="LaTeX:  \displaystyle 4 x^{2} " data-equation-content=" \displaystyle 4 x^{2} " /> , gives  <img class="equation_image" title=" \displaystyle 4 x^{2}(3 x - 4) " src="/equation_images/%20%5Cdisplaystyle%204%20x%5E%7B2%7D%283%20x%20-%204%29%20" alt="LaTeX:  \displaystyle 4 x^{2}(3 x - 4) " data-equation-content=" \displaystyle 4 x^{2}(3 x - 4) " /> . Grouping the last two terms and factoring out their GCF,  <img class="equation_image" title=" \displaystyle -5 " src="/equation_images/%20%5Cdisplaystyle%20-5%20" alt="LaTeX:  \displaystyle -5 " data-equation-content=" \displaystyle -5 " /> , gives  <img class="equation_image" title=" \displaystyle -5(3 x - 4) " src="/equation_images/%20%5Cdisplaystyle%20-5%283%20x%20-%204%29%20" alt="LaTeX:  \displaystyle -5(3 x - 4) " data-equation-content=" \displaystyle -5(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-%204%20" alt="LaTeX:  \displaystyle 3 x - 4 " data-equation-content=" \displaystyle 3 x - 4 " /> . This gives  <img class="equation_image" title=" \displaystyle 2[4 x^{2} \left(3 x - 4\right) -5 \cdot \left(3 x - 4\right)] = 2\left(3 x - 4\right) \left(4 x^{2} - 5\right) " src="/equation_images/%20%5Cdisplaystyle%202%5B4%20x%5E%7B2%7D%20%5Cleft%283%20x%20-%204%5Cright%29%20-5%20%5Ccdot%20%5Cleft%283%20x%20-%204%5Cright%29%5D%20%3D%202%5Cleft%283%20x%20-%204%5Cright%29%20%5Cleft%284%20x%5E%7B2%7D%20-%205%5Cright%29%20" alt="LaTeX:  \displaystyle 2[4 x^{2} \left(3 x - 4\right) -5 \cdot \left(3 x - 4\right)] = 2\left(3 x - 4\right) \left(4 x^{2} - 5\right) " data-equation-content=" \displaystyle 2[4 x^{2} \left(3 x - 4\right) -5 \cdot \left(3 x - 4\right)] = 2\left(3 x - 4\right) \left(4 x^{2} - 5\right) " /> . </p> </p>