Example of a tex file

example.tex

\include{preamble}
\subsection{Exercises}\label{p3-exercises}
\marginnote{p.32}
\marginnote{Notes to exercises}
\textbf{P3 vocabulary}
\begin{enumerate}
 \item For the polynomial $a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0$ the degree is \emph{n} and the leading coefficient is \emph{a}
 \item A polynomial that has all zero coefficients is called the \emph{zero polynomial}
 \item A polynomial with one term is called a \emph{monomial}
 \item the letters in FOIL stand for \emph{First}, \emph{Outer} \emph{Inner} \emph{Last}
 \item If a polynomial cannot be factored using integer coefficients, it is called \emph{prime}
 \item The polynomial $u^2 +2uv + v^2$ is called a \emph{trinomial}
\end{enumerate}
\begin{table}[ht]
    \caption{\textbf{Exercises 1-10}}
    \centering
    \begin{tabular}{clr}
        \toprule
        Exercise & Expresion & Example\\[5pt]
        \midrule
	    1 & A polynomial of degree zero & 7 \\[5pt]
	    2 & A trinomial of degree five & $-3x^5 + 2x^3 + x$ \\[5pt] 
	    3 & A binomial with a leading & \\
	    & coefficient 4 &  $1-4x^3$\\[5pt] 
	    4 & A monomial of positive degree & $6x$  \\[5pt]
	    5 & A trinomial with a leading & \\
	    & coefficient $\frac{3}{4}$ & $\frac{3}{4}x^4+x^2+14$ \\[5pt] 
	    6 & A third-degree polynomial & \\ 
	    & a with leading coefficient 1 & $x^3+2x^2-4x+1$\\[5pt] 
	    7 & A third-degree polynomial with & \\
	    & a leading coefficient -2 & $-2x^3+2x^2-2x$ \\[5pt] 
	    8 & A fifth-degree polynomial with & \\
	    & leading coefficient 8 & $8x^5 - 2x + 3$ \\[5pt]
	    9 & A fourth-degree polynomial with & \\
	    & a negative leading coefficient & $-4x^4 + 3x -2$ \\[5pt]
	    10 & A third-degree trinomial with & \\
	    & an even leading coefficient & $2x^3 + 3x^2 + 2$ \\[5pt] 
	    \bottomrule
	    \end{tabular}
\end{table} 
\begin{table}[ht]
    \caption{Exercises 11-16}
    \centering
    \begin{tabular}{crrcc}
        \toprule
        Exercise & Expression & Standard & Degree & Leading\\
		& & Form & & Coefficient\\[5pt]
		\midrule
		11 & $3x+4x^2+2$ & $4x^2 + 3x +2$ & 2 & 4 \\[5pt] 
		12 & $x^2-4-3x^4$ & $-3x^4 + x^2-4$ & 4 & -3 \\[5pt] 
		13 & $5-x^6$ & $-x^6+5$ & 5 & -1 \\[5pt] 
		14 & $-13 + x^2$ & $x^2 - 13$ & 2 & 1 \\[5pt] 
		15 & $1 -x + 6x^4 -2x^5$ & $-2x^5 + 6x^4 -x + 1$ & 5 & -2 \\[5pt] 
		16 & $7 + 8x$ & $8x+7$ & 1 & 8 \\[5pt] 
	    \bottomrule
	    \end{tabular}
\end{table} 
\clearpage
\marginnote{\\[25pt]
\noindent	
\textbf{Exercise 18} Last term has a negative integer exponent.\\
\noindent
\textbf{Exercise 19} Expression cannot be rewritten without a radical: $x\sqrt{(1+x)(1-x)}$}
\begin{table}[ht]
    \caption{Exercises 17-20}
    \centering
    \begin{tabular}{crcr}
        \toprule
        Exercise & Expression & Polynomial? & Standard Form\\[5pt]
        \midrule
	    17 & $3x + 4x^2 + 2$ & Y & $4x^2+3x+2$\\[5pt]
	    18 & $5x^4 -2x^2 + x^{-2}$ & N & \\[5pt] 
	    19 & $\sqrt{x^2-x^4}$ & N & \\[5pt] 
	    20 & $\frac{x^2+2x-3}{6}$ & Y & $\frac{1}{6}x^2+\frac{1}{3}x-\frac{1}{2}$\\[5pt] 
	    \bottomrule
	    \end{tabular}
\end{table} 

\begin{table}[ht]
    \caption{Exercises 21-36}
    \centering
    \begin{tabular}{crr}
        \toprule
        Exercise & Expression & Evaluated in\\
         &  & Standard Form\\[5pt]
        \midrule
	    21 & $(6x+5)-(8x+15)$ & $-2x -10$ \\[5pt]
	    22 & $(2x^2+1)-(x^2-2x+1)$ & $ x^2+2x$ \\[5pt] 
	    23 & $-(t^3-1)+(6t^3-5t)$ & $5t^3-5t+1 $ \\[5pt]
	    24 & $-(5x^2-1)-(-3x^2+5)$ & $ -2x^2-4$ \\[5pt]
	    25 & $(15x^2-6)-$& \\
	    & $(-8.1x^3-14.7x^2-17)$ & $ 8.1x^3 + 29.7x^2+11$ \\[5pt]
	    26 & $(15.6w-14w-17.4)-$ & \\
	    & $(16.9w^4-9.2w+13)$ & $-16.9w^4 + 10.8w -30.4 $ \\[5pt]
	    27 & $3x(x^2-2x+1)$ & $ 3x^2-6x^3+3x$ \\[5pt]
	    28 & $y^2(4y^2+2y-3)$ & $ 4y^4+2y^3-3y^2$ \\[5pt]
	    29 & $-5z(3z-1)$ & $ -15z^2+5z$ \\[5pt]
	    30 & $(-3x)(5x+2)$ & $-15x^2-6x $ \\[5pt]
	    31 & $(1-x^3)(4x)$ & $-4x^4+4x $ \\[5pt]
	    32 & $-4x(3-x^3)$ & $4x^4-12x $ \\[5pt]
	    33 & $(2.5x^2+5)(-3x)$ & $ -7.5x^3-15x$ \\[5pt]
	    34 & $(2-3.5y)(4y^3)$ & $ -14y^4+8y^3$ \\[5pt]
	    35 & $-2x(\frac{1}{8}x+3)$ & $-\frac{1}{4}x^2-6x $ \\[5pt]
	    36 & $6y(4-\frac{3}{8}y)$ & $-2\frac{1}{4}y^2+24y $ \\[5pt]
	    \bottomrule
	    \end{tabular}
\end{table}
\clearpage
\pagebreak
\marginnote{
\\
\noindent
Exercise 62: $[(x - 3y)+z][(x - 3y)+z]$ \\
\noindent
Let $a = (x - 3y) \\
(a + z)(a + z) = a2 +2az + z2 \\
a2  = (x - 3y)(x-3y) = x2 - 6xy + 9y2 \\
2az = 2z(x - 3y) = 2xz - 6yz \\
a^2 + 2az = x^2 - 6xy + 9y^2 + 2xz - 6yz + z^2$
} 
\begin{table}[ht]
    \caption{Exercises 37-68}
    \centering
    \begin{tabular}{crr}
        \toprule
        Exercise & Expression & Evaluation\\[5pt]
        \midrule
	     37 & $(x+3)(x+4)$ & $x^2+7x+12$\\[5pt]
	     38 & $(x-5)(x+10)$ & $x^2+5x-50$\\[5pt] 
	     39 & $(3x-5)(2x+1)$ & $6x^2 -7x -5$ \\[5pt] 
	     40 & $(7x-2)(4x-3)$ & $28x^2-29x+6$ \\[5pt] 
	     41 & $(2x-5y)^2$ & $x^2 - 20xy+25y^2$ \\[5pt] 
	     42 & $(5-8x)^2$ & $64x^2-80x+25$ \\[5pt] 
	     43 & $(x+10)(x-10)$ & $x^2-100$ \\[5pt]
	     44 & $(2x+3)(2x-3)$ & $4x^3-9$ \\[5pt] 
	     45 & $(x+2y)(x-2y)$ & $x^3-4y^2$ \\[5pt]
	     46 & $(4a+5b)(4a-5b)$ & $16a^2-25b^2$ \\[5pt] 
	     47 & $(2r^2-5)(2r^2+5)$ & $4r^2-25$ \\[5pt] 
	     48 & $(3a^3-4b^2)(3a^3+4b^2)$ & $9a^6-16b^4$ \\[5pt] 
	     49 & $(x+1)^3$ & $x^3 + 3x^2 + 3x +1$ \\[5pt] 
	     50 & $(y-4)^3$ & $y^3 - 48y^2 + 4y + 64$ \\[5pt]
	     51 & $(2x-y)^2$ & $4x^2-4xy + y^2$ \\[5pt] 
	     52 & $(3x+2y)^3$ & $27x^3+54x^2y + 36xy^2 + 8y^3$ \\[5pt] 
	     53 & $(\frac{1}{2}x-5)^2$ & $\frac{1}{4}x^2-5x + 25$ \\[5pt] 
	     54 & $(\frac{3}{5}t+4)^2$ & $\frac{9}{25}t^2 +8t + 16$ \\[5pt] 
	     55 & $(\frac{1}{4}x+3)(\frac{1}{4}x-3)$ & $\frac{1}{16}x^2 - 9$ \\[5pt]
	     56 & $(2x+\frac{1}{6})(2x-\frac{1}{6})$ & $4x^2 -\frac{1}{36}$ \\[5pt] 
	     57 & $(2.4x +3)^2$& $5.76x^2 + 14.4x + 9$\\[5pt] 
	     58 & $(1.8y-5)^2$ & $3.24y^2 - 18x + 25$ \\[5pt] 
	     59 & $(-x^2 +x -5)(3x^2+4x+1)$ & $-3x^4 -x^3 -12x^2 -19x -5$  \\[5pt] 
	     60 & $(x^2 + 3x + 2)(2x^2-x+4)$ & $2x^4 + 5x^3 +5x^2 +10x + 8$ \\[5pt]
	     61 & $[(x+2z) + 5][(x+2z)-5]$ & $x^2 + 4xz + 4z^2 -25$ \\[5pt] 
	     62 & $[(x-3y) + z][(x-3y) + z]$ & $x^2 - 6xy + 9y^2 + 2xz - 6yz + z^2
$ \\[5pt] 
	     63 & $[(x-3)+y]^2$ & $x^2 -6x + 2xy -6y + y^2 + 9$\\[5pt] 
	     64 & $[(x+1)-y]^2$ & $x^2 + 2x - 2xy - 2y + y^2  + 1$ \\[5pt] 
	     65 & $5x(x+1)-3x(x+1)$ & $2x^2 + 2x$ \\[5pt]  
	     66 & $(2x-1)(x+3)+3(x+3)$ & $2x^2 + 8x + 6$ \\[5pt] 
	     67 & $(u+2)(u-2)(u^2+4)$ & $u^4 - 16$ \\[5pt] 
	     68 & $(x+y)(x-y)(x^2+y^2)$ & $x^4 - y^4$ \\[5pt] 
	    \bottomrule
	    \end{tabular}
\end{table}

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