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\fancyhead[L]{CHAMP Geometry}
\fancyhead[C]{\textbf{{\Large Chapter 2 / Chapter 3}}}
\fancyhead[R]{Hw01}
\fancyfoot[L]{MSU}
\fancyfoot[C]{}
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\begin{document}
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%====================PAGE 1=======================================
\phantom{.}
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\textbf{Name :} \hrulefill\hrulefill\hrulefill \hspace{.2cm}
%\textbf{Date :} \hrulefill
\\
\vspace{.3cm}
\setlength{\fboxsep}{0pt}
\textbf{Instructions}\\[3pt]
\colorbox{gray!20}{\fbox{
\begin{minipage}{3.75in}\raggedright\small \vspace{4pt}\begin{itemize}
\item You are allowed to work with others to develop ideas on how to solve problems however when you write down your final solutions you should do so on your own to ensure your answers are your own.
\item Use the definitions, properties, strategies, and theorems in the book to help you solve your problems if you get stuck. Remember though that eventually (on quizzes and exams) you will want to be able to solve these problems without the book's assistance.
\end{itemize}\vspace{3pt} \end{minipage}
\begin{minipage}{3.75in}\raggedright\small\begin{itemize}
\item Leave your answer in exact form whenever possible.
\item {\bf Show all your work.} Write your answers clearly! Include enough steps for the grader to be able to follow your work. Don't skip equal signs, etc. Include words to clarify your reasoning.
\item It is recommended that you write a draft of your solution on scratch paper before writing you final solution on this sheet.
\setlength{\fboxsep}{2pt}
\item \fbox{BOX} your final answer (when applicable).
\end{itemize}\end{minipage} }}
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\setcounter{section}{2}
\begin{prob2}[1]
Find the remainder when $x^5=2x^3+x^2-4$ is divided by:
\begin{enumerate}[(a)]
\item $x-1$
\item $x+1$
\item $x-2$
\item $x+2$
\end{enumerate}
\end{prob2}
\begin{prob2}[5]
Find the quotient and the remainder when:
$(x^4-2x^3+5x+2)/(x+1)$
\end{prob2}
\begin{prob2}[10]
Find the quotient and the remainder when:
$(x^5+3x^2+4)/(x^2+2x+1)$
\end{prob2}
\begin{prob2}[12]
Determine whether $x-2$ or $x+2$ is a factor of $x^20-4x^18+3x-6$
\end{prob2}
\begin{prob2}[14]
Which of the following are factors of $P(x)=x^4-3x^3+5x-2$?
\begin{enumerate}[(a)]
\item $x+2$
\item $x-2$
\item $x+4$
\end{enumerate}
\end{prob2}
\begin{prob2}[19]
Find the remaining roots of $2x^3-5x^2-4x+3=0$, other than $x=3$.
\end{prob2}
\begin{prob2}[23]
Find the remaining roots of $x^4+3x^3-3x^2+3x-4=0$, $x=-4, x=1$
\end{prob2}
\setcounter{section}{3}
\begin{prob2}[7]
Sketch a graph of $y=x(1-x)(1+x)(2+x)$
\end{prob2}
\begin{prob2}[11]
Sketch a graph of $y=x^2(x+2)(x-1)(x+1)$
\end{prob2}
\begin{prob2}[15]
Factor each polynomial function and sketch its graph. $f(x)=x^4-x^2$
\end{prob2}
\begin{prob2}[20]
The graphs of $y=x^3$ and $y=x^5$ are shown at the right.
\includegraphics[width=2.5in]{pics/2-3-20.png}
%\includegraphics[width=2.5in]{pics/hw01-p01.png}
\begin{enumerate}[(a)]
\item Copy these graphs and then add the graph of $y=x^7$. (You may wish to use a computer or graphing calculator to check your graph)
\item What three points are common to all three graphs?
\end{enumerate}
\end{prob2}
\begin{prob2}[23]
Give an equation for each polynomial graph shown. More than one answer is possible.
\includegraphics[width=2.5in]{pics/2-3-23.png}
\end{prob2}
\begin{prob2}[37]
Find an equation of the cubic function whose graph passes through the points $(3, 0)$ and $(1,4)$ and is tangent to the x-axis at the origin.
\end{prob2}
\begin{prob2}[40]
If $P(x)$ is a cubic polynomial such that $P(0)=0, P(2)=-4$, and $P(x)$ is positive only when $x>4$ find $P(x)$.
\end{prob2}
\setcounter{section}{4}
\begin{prob2}[1]
A farmer wants to make a rectangular enclosure using a wall as one side and 120m of fencing for the other three sides, where x is the length of the side.
\begin{enumerate}[(a)]
\item Express the area in terms of x and state the domain of the area function.
\item Find the value of x that gives the greatest area.
\end{enumerate}
\end{prob2}
\begin{prob2}[4]
Suppose you have to use exactly 200m of fencing to make either one square enclosure or two separate square enclosures of any sizes you wish. What plan gives you the least area? the greatest area?
\end{prob2}
\begin{prob2}[9]
If a ball is thrown vertically upwards at 30 m/s, then its approximate height in meters $t$ seconds later is given by $h(t)=30t-5t^2$.
\begin{enumerate}[(a)]
\item After how many seconds does the ball hit the ground?
\item What is the domain of $h$?
\item How high does the ball go?
\end{enumerate}
\end{prob2}
\begin{prob2}[12]
An orange grower has 400 crates of fruit ready for market and will have 20 more for each day the grower waits. The present price is \$60 per crate and will drop an estimated \$2 per day for each day waited. In how many days should the grower ship the corp to maximize his income?
\end{prob2}
Cubic functions
\begin{prob2}[1]
A manufacturer cuts squares from the corners of an 8 cm by 14 cm piece of sheet metal and then folds the metal to make an open-top box.
\begin{enumerate}[(a)]
\item Show that the volume of the box is $V(x)=x(8-2x)(14-2x)$
\item What is the domain of $V$?
\item Find the approximate value of $x$ that maximizes the volume. Then give the approximate maximum volume.
\end{enumerate}
\end{prob2}
\begin{prob2}[9]
A cylinder is inscribed in a cone with height 10 and a base of radius 5, as shown at the right. Find the approximate values of $r$ and $h$ for which the volume of the cylinder is a maximum. Then give the approximate maximum volume.
\end{prob2}
\setcounter{section}{6}
\begin{prob2}[1]
Tell whether the equation is a polynomial equation that can be solved by grouping terms or a polynomial equation that has a quadratic form. Then solve the equation.
$x^4-4x^2-12=0$
\end{prob2}
\begin{prob2}[6]
Tell whether the equation is a polynomial equation that can be solved by grouping terms or a polynomial equation that has a quadratic form. Then solve the equation.
$2x^3-x^2-2x+1-0$
\end{prob2}
\begin{prob2}[17]
Use the rational root theorem to solve each equation giving all real and imaginary roots.
$3x^3-4x^2-5x+2=0$
\end{prob2}
\begin{prob2}[20]
Use the rational root theorem to solve each equation giving all real and imaginary roots.
$3x^3-x^2-36x+12=0$
\end{prob2}
\begin{prob2}[26]
Use the rational root theorem to solve each equation giving all real and imaginary roots.
$k(x)=-x^4+10x^2=24$
\end{prob2}
\begin{prob2}[35]
Sketch the graphs of the two given equations on a single set of axes. Then determine algebraically where the graphs intersect or are tangent.
$y=x^3+4x^2$, $y=3x+18$
\end{prob2}
\begin{prob2}[40]
\includegraphics[width=2.5in]{pics/2-6-40.png}
A cylinder is inscribed in a sphere of radius 4. Show that the volume of the cylinder is $V(x)=2\pi x(16-x^2)$. Find the two values of $x$ for which $V(x)=42\pi$.
\end{prob2}
\setcounter{section}{7}
Tell whether the statements in exercises 1-8 are true or false. Justify your answers.
\begin{prob2}[1]
Some cubic equations have no real roots.
\end{prob2}
\begin{prob2}[2]
Every polynomial equation has at least one real root.
\end{prob2}
\begin{prob2}[3]
The roots of a certain quartic equation are $\pm \frac{1}{2}$, $0,$ and $1+i$.
\end{prob2}
\begin{prob2}[4]
The roots of a certain fifth-degree equation are $-3, 4, 1-\sqrt{2}$, and $\pm i$
\end{prob2}
\begin{prob2}[5]
It is possible for the graph of a cubic function to be tangent to the x-axis at $x=-2, x=1,$ and $x=6$.
\end{prob2}
\begin{prob2}[6]
No polynomial equation can have an odd number of imaginary roots.
\end{prob2}
\begin{prob2}[7]
Suppose $P(x)$ is a polynomial with rational coefficients, and $b$ is rational but $\sqrt{b}$ is irrational. If $\sqrt{b}$ is a root of the equation $P(x)=0$, then $-\sqrt{b}$ is also a root.
\end{prob2}
\begin{prob2}[8]
If $a+bi$, $b \ne 0$, is a root of the polynomial equation $P(x)=0$, then the equation must have an even number of roots.
\end{prob2}
\begin{prob2}[10]
Find the sum and product of the roots of the given equation.
$6x^3-9x^2+x=0$.
\end{prob2}
\begin{prob2}[15]
Find a quadratic equation with integral coefficients that has the given roots.
$3 \pm \sqrt{2}$
\end{prob2}
\begin{prob2}[17]
A cubic equation with integral coefficients has no quadratic term. IF one root is $2+i\sqrt{5}$, what are the other roots?
\end{prob2}
\begin{prob2}[23]
Find a quadratic equation with integral coefficients that has roots $5-i\sqrt{3}$ and $i$.
\end{prob2}
\begin{prob2}[27]
Find an integer $c$ such that the equation $4x^3+cx-27=0$ has a double root.
\end{prob2}
\end{document}