in which the properties of operations on the real number system are recorded using symbols as "place holders" to denote constants and variables, and the rules governing mathematical expressions and equations involving these symbols are studied (note that this usually includes the subject matter of courses called intermediate algebra and college algebra), also called second year and third year 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 +, −, ×, ÷) 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 = 10"). 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) = 3x - 10, where f is the function, and x is the number to which the function is applied.").
A polynomial 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 variable is standardly denoted as exponentiation with a constant positive whole number exponent).
- For example, 2x+5 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 A related class of problems is finding algebraic expressions for the roots of a polynomial in a single variable.