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Problems PreCal 1508 PLTL Workshop, October 19, 2011\\
PLs: Alex Knaust, Edith Mejia. Lecturer: Yi-Yu Liao\\

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Please do the problems that you feel will help your group the most first (you don't have to do them in order). All handouts are available at \url{http://alex.knaust.info/pltlfall2011/}
\begin{enumerate}
\item Show that the following is true by combining the fractions on the left hand side.
\[ \frac{1}{x+1} + \frac{3}{x+2} - \frac{2}{(x+2)^2} = \frac{4x^2 + 11x + 8}{(x+1)(x+2)^2} \]

\item Write the form of the partial fraction decomposition for the following rational functions (do not solve for constants $A,B,...$)\\
\begin{center}
\begin{inparaenum}[a)]
	\item $\frac{x + 4}{x^2(3x -1)^2}\quad\quad\quad$
	\item $\frac{x^2}{x^4 - 2x^2 - 8}\quad\quad\quad$
	\item $\frac{x^4 + 2x^3 + 4x^2 + 8x + 2}{x^3 + 2x^2 + x}$
\end{inparaenum}
\end{center}

\item Write the partial fraction decomposition for the following rational functions (solve for constants)
\begin{center}
\begin{inparaenum}[a)]
	\item $\frac{3}{x^4 + x}\quad\quad\quad$
	\item $\frac{x^2 - 4x + 7}{(x+1)(x^2 - 2x + 3)}\quad\quad\quad$
	\item $\frac{x^2 - 4x}{x^2 + x + 6}$
\end{inparaenum}\end{center}

\item Perform either Gaussian Elimination with back-substitution or Gau\ss -Jordan elimination.\\

\begin{center}
\begin{inparaenum}[a)]
\item $\displaystyle \left\{\begin{array}{lr}
	-x + y = 4\\
	2x - 4y = -34\\
	\end{array}
	\right.\quad\quad\quad
$
\item $
\left[
\begin{array}{ccc|c}
1 & 0 & -3 & -2 \\ 
3 & 1 & -2 & 5 \\ 
2 & 2 & 1 & 4
\end{array}
\right]\quad\quad\quad
$
\item $\displaystyle \left\{\begin{array}{lr}
	3x + -2y + z = 15\\
	-x  + y +2z = -10\\
	x - y - 4z = 14	
	\end{array}
	\right.\quad\quad\quad
$
\end{inparaenum}
\end{center}

\item Swapping rows in an augmented matrix will not alter the solutions. True or False?

\item Describe the difference between a matrix in row-echelon form and a matrix in reduced-row-echelon form, give examples of matricies in both forms.

\item A polynomial, $f$, of degree 3 goes through the following points
\[ (-1, -5), (1, -1), (2, 1), (3, 11) \]
Find the function $f(x)$.

\item My peer leader told me that I can also do an analogue of partial fraction decomposition for rational numbers, using their prime factors as the denominators of the partial fractions, i.e. 
\[\frac{1}{18} = \frac{1}{2 \cdot 3^2} = \frac{A}{2} + \frac{B}{3} + \frac{C}{3^2}\]
Here he meant a fraction to be proper if the numerator is less than the denominator.
\begin{enumerate}
\item What is a potential decomposition of $\frac{4}{30}$?
\item Do you think he is right (Is this always possible)? 
\item Is the decomposition unique if there is one?
\end{enumerate}

\item When is a polynomial irreducible? Can you apply the same concept to say when an integer is irreducible?
\end{enumerate}



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