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Concept# Algebraic expression

Summary

In mathematics, an algebraic expression is an expression built up from constant algebraic numbers, variables, and the algebraic operations (addition, subtraction, multiplication, division and exponentiation by an exponent that is a rational number). For example, 3x2 − 2xy + c is an algebraic expression. Since taking the square root is the same as raising to the power 1/2, the following is also an algebraic expression:
An algebraic equation is an equation involving only algebraic expressions.
By contrast, transcendental numbers like pi and e are not algebraic, since they are not derived from integer constants and algebraic operations. Usually, pi is constructed as a geometric relationship, and the definition of e requires an infinite number of algebraic operations.
A rational expression is an expression that may be rewritten to a rational fraction by using the properties of the arithmetic operations (commutative properties and associative properties of addition and multiplication, distributive property and rules for the operations on the fractions). In other words, a rational expression is an expression which may be constructed from the variables and the constants by using only the four operations of arithmetic. Thus,
is a rational expression, whereas
is not.
A rational equation is an equation in which two rational fractions (or rational expressions) of the form
are set equal to each other. These expressions obey the same rules as fractions. The equations can be solved by cross-multiplying. Division by zero is undefined, so that a solution causing formal division by zero is rejected.
Algebra has its own terminology to describe parts of an expression:
1 – Exponent (power), 2 – coefficient, 3 – term, 4 – operator, 5 – constant, - variables
The roots of a polynomial expression of degree n, or equivalently the solutions of a polynomial equation, can always be written as algebraic expressions if n < 5 (see quadratic formula, cubic function, and quartic equation). Such a solution of an equation is called an algebraic solution.

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