Algebra is the brand of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures. Together with geometry, analysis, topology combinatorics, and number theory, algebra is one of the main branches of pure mathematics.
The part of algebra called elementary algebra is often part of the curriculum in secondary education and introduces the concept of variables representing numbers. Statements based on these variables are manipulated using the rules of operations that apply to numbers, such as addition.
This can be done for a variety of reasons, including equation solving. Algebra is much broader than elementary algebra and studies what happens when different rules of operations are used and when operations are devised for things other than numbers. Addition and multiplication can be generalized and their precise definitions lead to structures such as groups, rings and fields.
Elementary Algebra
Elementary algebra is the most basic form of algebra. It is taught to students who are presumed to have no knowledge of mathematics beyond the basic principles of arithmetic. In arithmetic, only numbers and their arithmetical operations (such as +, - , x ÷), occur. In algebra, numbers are often denoted by symbols (such as a.x or y). This is useful because it allows the general formulation of arithmetical laws (such as a+b=b+a for all a and b), and thus is the first step to a systematic exploration of the properties of the real number system.
It allows the reference to “unknown” numbers, the formulation of equations and the study of how to solve these (for instance, Find a number x such that 3x+1 = 10ยบ or going a bit further “Find a number x such that ax + b=c”. Step which lets to the conclusion that is not the nature of the specific numbers the one that allows us to solve it but that of the operations involved.
It allows the formulation of functional relationships (such as “If you sell x tickets, then your profit will be 3x-10 dollars, or f(x) = 4x – 10, where f is the function, and x is the number to which the function is applied.”).
Polynomial
A polynomial (see the article on polynomials for detail) is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, and multiplication (where repeated multiplication of the same variables is standardly denoted as exponentiation with a constant non-negative integer exponents) For example, x2 + 2x – 3 is a polynomial in the single variable x.
An important class of problems in algebra is factorization of polynomials, that is expressing a given polynomial as a product of other polynomials. The example polynomial above can be factored as (x –1) (x + 3). A related class of problems is finding a; gebraic expressions for the roots o a polynomial in a single variable.