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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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\chead{\LARGE  Worksheet 1 - Answers}
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\begin{document}


\begin{enumerate}
\item a) \ $x^{2}-25\qquad $b) \ $x^{2}-6x+9\qquad $c) $\ 4x^{2}-4x+1\qquad $%
d) \ $8x^{3}-12x^{2}+6x-1$

e) \ $27x^{3}-125\qquad $f) \ $-2x^{2}-8x+14$

\item a) \ $2$ \ \ \ \ \ b) \ $5$ \ \ \ \ c) \ $-7,\dfrac{1}{2},\dfrac{8}{5}$
\ \ \ \ \ \ d) \ $\dfrac{1}{2}$ \ \ \ \ \ \ e) \ $0,\dfrac{1}{2}$

\item a) \ $a^{4}\left( a+1\right) \qquad $b) \ $3\left( x-4\right) \left(
x+4\right) \qquad $c) \ $4a^{5}-4a$

\item a) \ $-2x\left( x+7\right) \left( x-9\right) \qquad $b) \ $4\left(
x+2\right) \left( x-8\right) \qquad $b) \ $x^{2}+2x+26\qquad $c) \ $-2\left(
x-3+\sqrt{7}\right) \left( x-3-\sqrt{7}\right) $

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

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

\item a) $\ 3-\sqrt{2},\sqrt{2}+3$.

b) \ Check $x=3-\sqrt{2}$ \ \ \ 
\begin{eqnarray*}
\text{LHS} &=&\left( 3-\sqrt{2}\right) ^{2}+7=9-6\sqrt{2}+2+7=18-6\sqrt{2} \\
\text{RHS} &=&6\left( 3-\sqrt{2}\right) =\allowbreak 18-6\sqrt{2}
\end{eqnarray*}%
Check $3+\sqrt{2}$ \ \ \ 
\begin{eqnarray*}
\text{LHS} &=&\left( 3+\sqrt{2}\right) ^{2}+7=9+6\sqrt{2}+2+7=18+6\sqrt{2} \\
\text{RHS} &=&6\left( 3+\sqrt{2}\right) =\allowbreak 18+6\sqrt{2}
\end{eqnarray*}%
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\item $\left( 6,-3\right) $

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\item Vertex: \ $\left( -4,-4\right) $ \ \ $\ x-$intercepts: \ $\left(
-6,0\right) $ and $\left( -2,0\right) $\newline
$y-$intercept: \ $\left( 0,12\right) $ \ \newline
additional points: $\left( -3,-3\right) ,$ $\left( -1,5\right) $

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\item $\sqrt{41}$ unit

\item $8\sqrt{21}\unit{cm}^{2}$

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

\item $49$ nickels and $23$ dimes

\item a) \ $1872$ feet\newline
b) \ $2$ seconds and $14$ seconds\newline
c) \ $19$ seconds\bigskip \bigskip 
\end{enumerate}

\end{document}