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Problems PreCal 1508 PLTL Workshop, September 16, 2011\\
PL: Alex Knaust, Lecturer: Yi-Yu Liao
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\begin{enumerate}
	\item For the polynomials below, determine if $a$ is a solution through 	synthetic division.
	\begin{enumerate}
		\item $f(x) = 3x^2 - 17x^2 + 15x - 25,\quad a=5$
	\end{enumerate}
	
	\item Is $(2x - 1)$ a factor of $6x^6 + x^5 -92x^4 +45x^3 + 			184x^2 + 4x - 48$?
		
	\item Are all real numbers complex numbers? Are all complex numbers real?
		
	\item For the following complex numbers, find $z \cdot w$ and $z + w$
	\begin{enumerate}
		\item $z = 2 + 3i, \quad w = -1 - \sqrt{-2}$
		\item $z = x + yi, \quad w = a + bi$
	\end{enumerate}
	
	\item Find all zeros (including complex ones) of $f(x) = x^2 + x + 2$
	
	\item If $z$ is a complex number, $\bar{z}$ often denotes its conjugate. If $w = 3 - 7i$ what is $\bar{w}$? $w \cdot \bar{w}$?
	
	\item Write $\frac{2 + 3i}{i - 1}$ in \emph{standard form ($a + bi$)}.
			
	\item What is $i^3$? $i^{20}$? $i^n$ for a positive integer $n$?
	
	
	\item If $z \in \mathbb{C}$ is a root of $p(x) = ax^2 + bx + c$, is $\bar{z}$ also a root? \emph{Hint : think of the quadratic formula}

	
\end{enumerate}
\begin{notation} The following symbols are often used to denote certain sets.
$\mathbb{C}$ = Complex numbers, $\mathbb{R}$ = Real numbers, $\mathbb{N}$ = Natural numbers (Integers greater than 0), $\mathbb{Z}$ = Integers, $\mathbb{Q}$ = Rational numbers (fractions with numerator and denominator in $\mathbb{Z}$
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