# Algebra (elementary mathematics)

The elementary algebra east is a branch of the Mathématiques whose object is the study of the laws which govern the numerical quantities. The qualifier of elementary appears at the same time as the modern algebra in order to differentiate it from this one. Today, it is the first approach of the algebra in the school course.

The algebra is different from the arithmetic by the introduction of letters ($a$, $b$, $c$,…, $x$, $y$, $z$,…, $\ alpha$, $\ beta$, $\ gamma$,…) indifferently representing all the numbers and to which are applied same the rules of calculations that if it were about numbers.

It is thus possible to establish laws depending only on the nature of the operations, independently of the numbers.

The resolutions of equations and inequations, the study of the polynomials are applications of the algebra.

### Algebraical expressions

An algebraical expression consists of numbers, letters and operational signs:

• the sign $+$ is used to mark the Addition.
• the sign $-$ is used to mark the Soustraction.
• the signs $\ times$ or $\ cdot$ are used to mark the Multiplication. When the multiplication relates to two letters, it possible to write $ab$ instead of $a \ times b$.
• the sign $\ div$ is used to mark the Division, $a \ div b$ also being able to be written $\ cfrac \left\{has\right\} \left\{B\right\}$.

For example:

• the product of a $x$ number increased of 3 by itself is written $\left(x+3\right) x$.
• the difference of the squares of two $a$ numbers and $b$ is written $a^2-b^2$

Évaluer an algebraical expression consists in allotting a value to each unknown factor, then to carry out arithmetic calculation obtained.

For example to evaluate the expression $x^2+x-1$ for $x=2$ consists in carrying out calculation $2^2+2-1$.

### Factorization and development

To factorize an algebraical expression, $E$, consists in transforming of it the writing in the shape of a product of two or several expressions ($A$, $B$,…) :

$E = has \ times B \ times \ cdots$
Each one of the expressions $A$, $B$,… is called a factor.

To develop an algebraical expression, $E$, consists in transforming of it the writing in the form of a sum (or difference) of two or several expressions. ($A$, $B$,…) :

$E = has + B + \ cdots$
Each one of the expressions $A$, $B$,… is called a term.

## Algebraic example of resolution of a problem

Simple problem: 2 tickets are worth 100 euros. What is worth a ticket? We can disregard nature of the ticket (euro, rouble, dollar,…), the resolution of the problem will be always the same one.

By noting $x$ the value of a ticket, the problem results in the equation $2x=100$.

By dividing each member by 2, one obtains $x=50$.

## See too

### To deepen

Simple: Elementary will algebra

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