# Algebraic expression

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:

${\sqrt {\frac {1-x^{2}}{1+x^{2}}}}$ An algebraic equation is an equation involving only algebraic expressions.

By contrast, transcendental numbers like π and e are not algebraic, since they are not derived from integer constants and algebraic operations. Usually, π 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,

${\frac {3x-2xy+c}{y-1}}$ is a rational expression, whereas

${\sqrt {\frac {1-x^{2}}{1+x^{2}}}}$ is not.

A rational equation is an equation in which two rational fractions (or rational expressions) of the form

${\frac {P(x)}{Q(x)}}$ 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.

## Terminology

Algebra has its own terminology to describe parts of an expression: 1 – Exponent (power), 2 – coefficient, 3 – term, 4 – operator, 5 – constant, $x,y$ - variables

## In roots of polynomials

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. But the Abel–Ruffini theorem states that algebraic solutions do not exist for all such equations (just for some of them) if n $\geq$ 5.

## Conventions

### Variables

By convention, letters at the beginning of the alphabet (e.g. $a,b,c$ ) are typically used to represent constants, and those toward the end of the alphabet (e.g. $x,y$ and $z$ ) are used to represent variables. They are usually written in italics.

### Exponents

By convention, terms with the highest power (exponent), are written on the left, for example, $x^{2}$ is written to the left of $x$ . When a coefficient is one, it is usually omitted (e.g. $1x^{2}$ is written $x^{2}$ ). Likewise when the exponent (power) is one, (e.g. $3x^{1}$ is written $3x$ ), and, when the exponent is zero, the result is always 1 (e.g. $3x^{0}$ is written $3$ , since $x^{0}$ is always $1$ ).

## Algebraic and other mathematical expressions

The table below summarizes how algebraic expressions compare with several other types of mathematical expressions by the type of elements they may contain, according to common but not universal conventions.

Arithmetic expressions Polynomial expressions Algebraic expressions Closed-form expressions Analytic expressions Mathematical expressions
Constant Yes Yes Yes Yes Yes Yes
Elementary arithmetic operation Yes Addition, subtraction, and multiplication only Yes Yes Yes Yes
Finite sum Yes Yes Yes Yes Yes Yes
Finite product Yes Yes Yes Yes Yes Yes
Finite continued fraction Yes No Yes Yes Yes Yes
Variable No Yes Yes Yes Yes Yes
Integer exponent No Yes Yes Yes Yes Yes
Integer nth root No No Yes Yes Yes Yes
Rational exponent No No Yes Yes Yes Yes
Integer factorial No No Yes Yes Yes Yes
Irrational exponent No No No Yes Yes Yes
Exponential function No No No Yes Yes Yes
Logarithm No No No Yes Yes Yes
Trigonometric function No No No Yes Yes Yes
Inverse trigonometric function No No No Yes Yes Yes
Hyperbolic function No No No Yes Yes Yes
Inverse hyperbolic function No No No Yes Yes Yes
Root of a polynomial that is not an algebraic solution No No No No Yes Yes
Gamma function and factorial of a non-integer No No No No Yes Yes
Bessel function No No No No Yes Yes
Special function No No No No Yes Yes
Infinite sum (series) (including power series) No No No No Convergent only Yes
Infinite product No No No No Convergent only Yes
Infinite continued fraction No No No No Convergent only Yes
Limit No No No No No Yes
Derivative No No No No No Yes
Integral No No No No No Yes

A rational algebraic expression (or rational expression) is an algebraic expression that can be written as a quotient of polynomials, such as x2 + 4x + 4. An irrational algebraic expression is one that is not rational, such as x + 4.