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\newtheorem{theorem}{Theorem}
\newtheorem{acknowledgement}[theorem]{Acknowledgement}
\newtheorem{algorithm}[theorem]{Algorithm}
\newtheorem{axiom}[theorem]{Axiom}
\newtheorem{case}[theorem]{Case}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{conclusion}[theorem]{Conclusion}
\newtheorem{condition}[theorem]{Condition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{criterion}[theorem]{Criterion}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{exercise}[theorem]{Exercise}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{solution}[theorem]{Solution}
\newtheorem{summary}[theorem]{Summary}
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\begin{document}


\begin{enumerate}
\item Expand each of the following.

a) \ $\left( x+5\right) \left( x-5\right) =x^{2}-25$

b) \ $\left( x-3\right) ^{2}=x^{2}-6x+9$

c) $\ \left( 2x-1\right) ^{2}=4x^{2}-4x+1$

d) \ $\left( 2x-1\right) ^{3}=8x^{3}-12x^{2}+6x-1$

e) \ $\left( 3x-5\right) \left( 9x^{2}+15x+25\right) =27x^{3}-125$

\item Solve each of the following equations. \ Make sure to check your
solution(s).

a) \ $\left( 2x-1\right) ^{2}-3x\left( x-2\right) =8+\left( x-3\right) ^{2}$
\ \ \ \ \ \ \ \ \ \ $2$

b) \ $\dfrac{3x+1}{4}-\dfrac{2x-1}{3}=x-4$ \ \ \ \ \ \ \ \ \ \ \ \ \ $\ \ 5$

c) \ $\left( x+7\right) \left( 2x-1\right) \left( 5x-8\right) =0$ \ \ \ \ \
\ \ \ \ \ \ \ $-7,\dfrac{1}{2},\dfrac{8}{5}$

d) \ $\left( 2x-1\right) ^{5}=0$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $%
\dfrac{1}{2}$

e) \ $6x^{3}=3x^{2}$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $\ \ \ \ \ \ \ \ 0,%
\dfrac{1}{2}$

\item Completely factor each of the following over the real numbers.

a) \ $a^{5}+a^{4}=a^{4}\left( a+1\right) $

b) \ $3x^{2}-48=3\left( x-4\right) \left( x+4\right) $

c) \ $4a^{5}-4a=4a\left( a^{2}+1\right) \left( a-1\right) \left( a+1\right) $

\item Factor each of the following over the real numbers by completing the
square.

a) \ $126x+4x^{2}-2x^{3}=-2x\left( x+7\right) \left( x-9\right) $

b) \ $4x^{2}-24x-64=4\left( x+2\right) \left( x-8\right) $

b) \ $2x+x^{2}+26=\left( x+1\right) ^{2}+25$

c) \ $12x-2x^{2}-4=-2\left( x-3+\sqrt{7}\right) \left( x-3-\sqrt{7}\right) $

d) \ $2x^{2}+x-15=2\left( x+3\right) \left( x-\dfrac{5}{2}\right) $

\item Simplify each of the following expressions.

a) \ $\sqrt{50}-3\sqrt{18}+\sqrt{200}$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ c) \ Rationalize $\ \dfrac{3}{\sqrt{10}-4}$

b) \ $\left( 8-\sqrt{5}\right) ^{2}$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ d) \ Find the exact value of \ $x^{2}-4x+7$ \ if \ 
$x=5-\sqrt{2}$

a) \ $6\sqrt{2}$ \ \ \ \ \ \ b) \ $69-16\sqrt{5}$ \ \ \ \ \ c) \ $-2-\dfrac{1%
}{2}\sqrt{10}$\ \ \ \ \ \ \ \ d) \ $14-6\sqrt{2}$

\item a) \ Solve the equation $x^{2}+7=6x$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $3-%
\sqrt{2},\sqrt{2}+3$.

b) \ Use exact values to check your solutions.

\item Graph the straight lines $y=-\dfrac{2}{3}x+1$ and $x+2y=0$ in the same
coordinate system. Use your graph to find the coordinates of the point where
the lines intersect.

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\item Graph the parabola $y=8x+x^{2}+12$. Clearly label the coordinates of
five points on the parabola, including vertex and intercepts.

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\item Find the distance between $\left( 7,-2\right) $ \ and $\left(
3,3\right) $. \ \ \ \ \ \ \ \ \ $\sqrt{41}$

\item Compute the exact value of the area of the triangle with sides $10%
\unit{cm},$ $10\unit{cm},$ and $8\unit{cm}$ long.

\item Write an equation for the line that is\newline
a) \ parallel to $2x-5y=17$ and passes through $P\left( 10,-2\right) $.%
\newline
b) \ perpendicular to $2x-5y=17$ and passes through $P\left( 10,-2\right) $.%
\newline
c) \ passes through the points $\left( -2,4\right) $ and $\left( 6,0\right) $%
.

a) \ $y=\dfrac{2}{5}x-6$ \ \ \ \ \ \ \ b) \ $y=-\dfrac{5}{2}x+23$ \ \ \ \ \
\ c) $\ y=-\dfrac{1}{2}x+3$\smallskip 

\item We have $72$ coins in a jar, all dimes and nickels. \ How many of each
coin do we have if the totalo value of all coins is $\$4.75$?

$49$ nickels and $23$ dimes

\item We are standing on the top of a $912$ feet tall tower and throw a
small object upward. \ We measure and record the vertical location of the
object. \ (Location is measured in feet, time in seconds.) \ $t$ seconds
after we threw it, the vertical position of the object is%
\begin{equation*}
h\left( t\right) =-16t^{2}+256t+912
\end{equation*}

a) \ Where is the object $6$ seconds after we threw it? \ \ \ $1872$\newline
b) \ How long does it take for the object to be at a height of $1360$ feet
above the ground? \ \ \ \ $2,14$\newline
c) \ How long does it take until the obejct hits the ground? \ \ \ \ $19$%
s\bigskip \bigskip 
\end{enumerate}

\end{document}