yzengmath1

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\section{How To Simplify Fractions to the Lowest Term \\ Ye Zeng}

\centerline{\includegraphics[scale=0.8]{yzengimg1.jpg}}

%Please list key words here.  Use a new \item for each word.  Please provide a short definition of each key word. 
\subsection{Key Words}

\begin{itemize}
\item {\bf Polynomials} -  An expression that can have constants, variables and exponents. \\
	E.g. $5xy^2 - 3x + 5y^3 - 3$
\item {\bf Lowest term} - The numerator and denominator of a fraction have no common factor except number one.\\
    E.g. $\frac{3}{5x}$
\item {\bf Numerator} - The top part of a fraction.
\item {\bf Denominator} - The bottom part of a fraction.
\end{itemize}

%Please put your sample problem below. It should sound like a test question and should be taken from a sample test, quiz or homework. 
\subsection{Sample Question}

Which of the following shows the expression $\frac{3x}{10x+x^2}$ reduced to the lowest terms?

A. $\frac{3x}{10+x}$

B. $\frac{3}{10+x}$

C. $\frac{1}{7+x}$

D. $\frac{3}{10x}$



%Please show all work in your sample solution.  Use colors where appropriate. Do not have too much information on one line (no more than 2 equal signs on any given line). Make sure your solution is correct
\subsection{Solution}

The answer is {\bf B}.

{\bf Steps to solve this question}

\begin{enumerate}
\item Combine like-terms.\\

$\frac{3x}{x(10+x)}$
\item Cancel x (the common factor) in both numerator and denominator.\\

$\frac{3}{10+x}$


\end{enumerate}


%You should have one step for each equal sign in your solution. Please use appropriate vocabulary and be as specific as possible.  Again one \item for each step. 
\subsection{Steps to Solve This Kind of Problem}

\begin{enumerate}
\item Always combine like-terms first.
\item Canceled if there is a common factor in both numerator and denominator.
\item If there's no common factor, that's the lowest term.
\end{enumerate}

\subsection{Challange Question}
{\bf Challenge Question}

\begin{center}
$\frac{3(x+1)}{x^2-1}$\\
\end{center}

{\bf Solution}

\begin{enumerate}
\item Factorization\\

$\frac{3(x+1)}{(x+1)(x-1)}$\\

\item Cancel $(x+1)$ in both numerator and denominator.\\

$\frac{3}{x-1}$


\end{enumerate}


%Please add any other notes or important things to remember when solving this type of problem. 
\subsection{Notes/other things to remember}

\begin{itemize}
\item $\frac{a^m}{a^n}$ = $a^{m - n}$
\item $({a^m})^{n}$ = ${a}^{mn}$
\item $\frac{am - an}{a}$

= $\frac{a(m - n)}{a}$ 

= $m - n$
\item $(x+y)^2$ = $x^2 + 2xy + y^2$
\item $(x-y)^2$ = $x^2 - 2xy + y^2$
\item $(x^2 - 1)$ = $(x+1)(x-1)$
\item $x^3-1$ = $(x-1)(x^2+x+1)$
\item $x^3+1$ = $(x+1)(x^2-x+1)$
\end{itemize}


\end{document}