\chapter{Algebraic equations}


 We've created a notation for variables and define the field of real numbers; it's time to bring to fruition the power these tools give mathematicians to solve problems.

Notably, elementary algebra allows the existence for algebraic expressions and equations.

\section{Algebraic expressions and substitution}

\begin{definition}
an \emph{algebraic expression} is an equation containing constants and variables connected by the 4 operations and integer exponentiation.
\end{definition}

- give examples of algebraic expressions

\begin{example}
$6x+2$
$x^2-3x$
$\frac{x-1}{x+1}$
\end{example}

consider a algebraic expression that 

- describe method of substitution
\subsection{Substitution into algebraic expression}

In algebraic expressions, a variable is acting as a 'placeholder' for a number; one can 'substitute' a number with the variable and then calculate the expression's value.

We then use BODMAS and FOIL to calculate the final value.
\begin{example}
$6x-4,x=2$
$6(2)-4$
$12-4$
$8$
\end{example}


\section{Algebraic equations and their solutions}


We'll consider some basic algebraic equations and study basic techniques that can be used to solve them.
\begin{definition}
an \emph{algebraic equation} is an equation that equates algebraic expressions to one another.
\end{definition}


\begin{example}
$6x+2=32$
$x^2-3x=0$
$\frac{x-1}{x+1}=x$
\end{example}


- give examples of algebraic equations and their solutions
- describe method of algebraic manipulation




\section{Algebraic identities}


These identities describe the more deeper properties of real numbers, and they are indeed useful in the study of polynomial algebraic equations (we'll come to this soon).
\begin{theorem}[Difference of two squares]
Let $x,y$ be real numbers, then
\[ x^2 -y^2=(x+y)(x-y) ]\
%\[ x,y \in \mathhbb{R} \implies x^2-y^2=(x+y)(x-y) \]
\end{theorem}

This can be generalized to the following.

\begin{theorem}[Difference of two powers]
Let $x,y$ be real numbers and $n$ br a natural number, then
\[ x^n -y^n=(x-y)(x^{n-1} + x^{n-2}y + \ldots + xy^{n-2}+ y^{n-1} ]\
%	\[ x,y \in \mathhbb{R} \land n \in \mathbb{N} \implies x^n-y^n=(x-y)(\sum^{n-1}_{i=0}x^{n-1-i}y^{i}) \]
\end{theorem}

The square of the sum of two numbers can be broken down as such due to the FOIL method.

\begin{theorem}
Let $x,y$ be real numbers, then
\[ (x+y)^2=x^2+2xy+y^2 ]\
\[ x,y \in \mathhbb{R} \implies (x+y)^2=x^2+2xy+y^2 \]
\end{theorem}


\begin{theorem}
Let $x,y$ be real numbers, then
\[ (x+a)(x+b)= x^2 + (a+b)x+ab \]
\[ x,y \in \mathhbb{R} \implies (x+a)(x+b)= x^2 + (a+b)x+ab \]
\end{theorem}


One may notice that similar methods lead to an identity for $(x+y)^3$, is there an algebraic identity that generalizes to $(x+y)^n$ for any nonnegative integer $n$? The binomial theorem is precisely this, and it will be revealed towards the end of this book.





\section{Subtle points in algebra}

$x^{2n}=y$
$x^{2n} = \pm \sqrt[2n]{y}$


$\sin (x) = y$

$x=\sin^{-1}(y)+2 \pi k$