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\begin{center}\huge MATH 1101 - EXAM 1 SPRING 2017\\
SOLUTION
\end{center}

\begin{flushleft}\vspace{0.5in}
Thursday 9 February 2017 \\
Instructor: Tom Cuchta
\end{flushleft}

\textbf{Instructions:}
\begin{itemize}
	\item  Show all work, clearly and in order, if you want to get full
	credit. If you claim something is true \textbf{you must show work backing up your claim}. I reserve the right to take off points if I cannot see how you arrived at your answer (even if your final answer is correct).
	\item Justify your answers algebraically whenever possible to ensure full credit. 	
	\item  Circle or otherwise indicate your final answers.
	\item  Please keep your written answers brief; be clear and to the point.
	\item  Good luck!
\end{itemize}
\vspace*{10pt}


\end{coverpages}

\begin{questions}
\question[10] List all that apply: natural number, integer, rational number, irrational number.
\begin{parts}
\part[2] $\dfrac{3}{2}$ \\
\textit{Solution:} rational number
\part[2] $23$ \\
\textit{Solution:} natural number, integer, rational number
\part[2] $\sqrt{5}$ \\
\textit{Solution:} irrational number
\part[2] $0.222\ldots$ \\
\textit{Solution:} rational number
\part[2] $\sqrt{25}$ \\
\textit{Solution:} natural number, integer, rational number
\end{parts}

\question[10] Simplify.
\begin{parts}
\part[2] $3+8 \cdot 2$ \\
\textit{Solution:} Calculate
$$3+8\cdot 2=3+16=19.$$
\part[2] $|5-3| + |2-7|$ \\
\textit{Solution:} Calculate
$$|5-3|+|2-7|=|2|+|-5|=2+5=7.$$
\part[3] $2+6\div 2+1$ \\
\textit{Solution:} Calculate
$$2+6 \div 2 +1 = 2+3+1=6.$$
\part[3] $3 \cdot 4 + [3-(2\cdot 4-3)]^2$ \\
\textit{Solution:} 
$$\begin{array}{ll}
3 \cdot 4 + [3-(2\cdot 4-3)]^2 &= 12 + [3-(8-3)]^2 \\
&=12 + [3-5]^2 \\
&= 12 +(-2)^2 \\
&= 12 + 4 \\
&= 16.
\end{array}$$
\end{parts}

\question[10] Simplify using the rules of exponents. Write the final answer using only positive exponents.
\begin{parts}
\part[3] $x^2 x^3$ \\
\textit{Solution:} Calculate
$$x^2 x^3 = x^{2+3}=x^5.$$
\part[3] $\left(\dfrac{x}{3y}\right)^2$ \\
\textit{Solution:} Calculate
$$\begin{array}{ll}
\left( \dfrac{x}{3y} \right) &= \dfrac{x^2}{(3y)^2} \\
&= \dfrac{x^2}{9y^2}.
\end{array}$$
\part[4] $\left( \dfrac{x}{y} \right)^4 \left( \dfrac{y^2}{3x^3} \right)$ \\
\textit{Solution:} Calculate
$$\begin{array}{ll}
\left( \dfrac{x}{y} \right)^4 \left( \dfrac{y^2}{3x^3} \right) &= \dfrac{x^4}{y^4} \left( \dfrac{y^2}{3x^3} \right)  \\
&= \dfrac{1}{3} x^{4-3} y^{2-4} \\
&=\dfrac{1}{3} x y^{-2} \\
&= \dfrac{x}{3y^2}.
\end{array}$$
\end{parts}
\question[10] Simplify.
\begin{parts}
\part[2] $\sqrt{4}$ \\
\textit{Solution:} Since $4=2 \cdot 2$ we see
$$\sqrt{4}=\sqrt{2 \cdot 2} = 2.$$
\part[4] $\sqrt[3]{-8}$\\
\textit{Solution:} Since $-8=(-2)(-2)(-2)$ we see
$$\sqrt[3]{-8} = \sqrt[3]{(-2)(-2)(-2)}=-2.$$
\part[4] $\sqrt[4]{81}$\\
\textit{Solution:}  Since $81=3 \cdot 3 \cdot 3 \cdot 3$ we see
$$\sqrt[4]{81}=\sqrt[4]{3 \cdot 3 \cdot 3 \cdot 3}=3.$$
\end{parts}

\question[15] Collect like terms.
\begin{parts}
\part[5] $3x+5y-2x+y$ \\
\textit{Solution:} Calculate
$$3x+5y-2x+y=x+6y.$$
\part[5] $2(x+1) + 3(x+2)$\\
\textit{Solution:} Calculate
$$2(x+1)+3(x+2)=(2x+2)+(3x+6)=5x+8.$$
\part[5] $2\{3x^2 + 3[2x-(5-x)]\}$ \\
\textit{Solution:} Calculate 
$$\begin{array}{ll}
2 \{ 3x^2 + 3[2x-(5-x)]\} &= 2 \{ 3x^2 + 3[2x-5+x]\} \\
&=2 \{ 3x^2 + 3[3x-5]\} \\
&=2\{ 3x^2 + 9x - 15 \} \\
&= 6x^2+18x-30.
\end{array}$$
\end{parts}

\question[5] Write an algebraic expression and simplify: 
\begin{center}``the total amount of money (in cents) of $x+2$ nickels, $x-1$ dimes, and $x$ quarters"\end{center} 
\textit{note: recall that nickels are worth $5$ cents, dimes are worth $10$ cents, and quarters are worth $25$ cents}  \\
\textit{Solution:} Write
$$5(x+2)+10(x-1)+25x =(5x+10)+(10x-10)+25x=40x.$$
\question[15] Multiply and simplify. 
\begin{parts}
\part[5] $x(x+2)+x$ \\
\textit{Solution:} Calculate
$$x(x+2)+x=x^2+2x+x=x^2+3x.$$
\part[5] $(x+3)(x-2)$ \\
\textit{Solution:} Calculate
$$\begin{array}{ll}
(x+3)(x-2) &= (x+3)(x)-(x+3)(2) \\
&= (x^2+3x)-(2x+6) \\
&= x^2+x-6.
\end{array}$$
\part[5] $(2x+3)(x^2+2x+1)$ \\
\textit{Solution:} Calculate
$$\begin{array}{ll}
(2x+3)(x^2+2x+1) &= (2x+3)(x^2) + (2x+3)(2x) + (2x+3)(1) \\
&= (2x^3+3x^2)+(4x^2+6x) + (2x+3) \\
&=2x^3 + 7x^2+8x+3.
\end{array}$$
\end{parts}

\question[15] Solve the equation.
\begin{parts}
\part[5] $x+3=4$ \\
\textit{Solution:} Add $-3$ to both sides 
$$x+3 + -3 = 4 + (-3),$$
and simplify to get
$$x=1.$$
\part[5] $2(x+5)=15$ \\
\textit{Solution:} First distribute the $2$ into the sum to get
$$2x+10=15.$$
Now add $(-10)$ to both sides to get
$$2x=5.$$
Now multiply both sides by $\dfrac{1}{2}$ to get
$$x = \dfrac{5}{2}.$$
\part[5] $3(x+1)=4(x+2)$ \\
\textit{Solution:}First multiply the $3$ through the sum on the left and multiply the $4$ through the sum on the right to get
$$3x+3=4x+8.$$
Add $(-3x)$ to both sides to get
$$3=x+8.$$
Add $(-8)$ to get
$$3-8=x,$$
or
$$-5=x.$$
\end{parts}
\question[10] Perform the polynomial long division $(x^2+9x+14) \div (x+7)$. \\
\textit{Solution:} Calculate
$$\polylongdiv{x^2+9x+14}{x+7}$$
\vfill \vfill
\end{questions}

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