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In mathematics, an algebraic expression is an expression built up from integer constants, variables, and the algebraic operations (addition, subtraction, multiplication, division and exponentiation by an exponent that is a rational number).[1] For example, $$3x^2 - 2xy + c$$ is an algebraic expression. Since taking the square root is the same as raising to the power $$\tfrac{1}{2}$$,

$$\sqrt{\frac{1-x^2}{1+x^2}}$$

is also an algebraic expression. By contrast, transcendental numbers like π and e are not algebraic.

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, $$\frac{3x^2 - 2xy + c}{y^3-1}$$ 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:

Algebraic equation notation

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 $$n \ge 5$$ .
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.[2] They are usually written in italics.[3]
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 $$x^2)$$ .[4] Likewise when the exponent (power) is one, (e.g. 3x^1 is written 3x),[5] and, when the exponent is zero, the result is always 1 (e.g. $$3x^0$$is written 3, since $$x^0$$ is always 1).[6]
Algebraic vs. other mathematical expressions

The table below summarizes how algebraic expressions compare with several other types of mathematical expressions.

Arithmetic expressions Polynomial expression Algebraic expressions Closed-form expressions Analytical expressions Mathematical expressions
Constant Yes Yes Yes Yes Yes Yes
Variable Yes Yes Yes Yes Yes Yes
Elementary arithmetic operation Yes Yes Yes Yes Yes Yes
Factorial Yes Yes Yes Yes Yes Yes
Integer exponent No Yes Yes Yes Yes Yes
N-th root No No Yes Yes Yes Yes
Rational exponent No No Yes Yes Yes Yes
Irrational exponent 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
Gamma function No No No No Yes Yes
Bessel function No No No No Yes Yes
Special function No No No No Yes Yes
Continued fraction No No No No Yes Yes
Infinite series No No No No Yes Yes
Formal power series No No No No No Yes
Differential No No No No No Yes
Limit 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.

Algebraic equation
Linear_equation#Algebraic_equations
Algebraic function
Analytical expression
Arithmetic expression
Closed-form expression
Expression (mathematics)
Polynomial
Term (logic)

Notes

Morris, Christopher G. (1992). Academic Press dictionary of science and technology. p. 74.
William L. Hosch (editor), The Britannica Guide to Algebra and Trigonometry, Britannica Educational Publishing, The Rosen Publishing Group, 2010, ISBN 1615302190, 9781615302192, page 71
James E. Gentle, Numerical Linear Algebra for Applications in Statistics, Publisher: Springer, 1998, ISBN 0387985425, 9780387985428, 221 pages, [James E. Gentle page 183]
David Alan Herzog, Teach Yourself Visually Algebra, Publisher John Wiley & Sons, 2008, ISBN 0470185597, 9780470185599, 304 pages, page 72
John C. Peterson, Technical Mathematics With Calculus, Publisher Cengage Learning, 2003, ISBN 0766861899, 9780766861893, 1613 pages, page 31

Jerome E. Kaufmann, Karen L. Schwitters, Algebra for College Students, Publisher Cengage Learning, 2010, ISBN 0538733543, 9780538733540, 803 pages, page 222

References

James, Robert Clarke; James, Glenn (1992). Mathematics dictionary. p. 8.