Algebraic expression

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

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).

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 \ge 5.

Rational expressions

Given two polynomials and , their quotient is called a rational expression or simply rational fraction.{{cite book

\frac{x^3+x^2+1}{x^2-5x+6} = (x+6) + \frac{24x-35}{x^2-5x+6},

where the second term is a proper rational fraction. The sum of two proper rational fractions is a proper rational fraction as well. The reverse process of expressing a proper rational fraction as the sum of two or more fractions is called resolving it into partial fractions. For example,

\frac{2x}{x^2-1} = \frac{1}{x-1} + \frac{1}{x+1}.

Here, the two terms on the right are called partial fractions.

Irrational fraction

An irrational fraction is one that contains the variable under a fractional exponent.{{cite book

\frac{x^{1/2} - \tfrac13 a}{x^{1/3} - x^{1/2}}.

The process of transforming an irrational fraction to a rational fraction is known as rationalization. Every irrational fraction in which the radicals are monomials may be rationalized by finding the least common multiple of the indices of the roots, and substituting the variable for another variable with the least common multiple as exponent. In the example given, the least common multiple is 6, hence we can substitute x = z^6 to obtain

\frac{z^3 - \tfrac13 a}{z^2 - z^3}.

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.

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 .

Notes

References

References

1. [http://www.daviddarling.info/encyclopedia/A/algebraic_function.html Definition of "Algebraic function"] {{Webarchive. link. (2020-10-26 in [[David J. Darling]]'s Internet Encyclopedia of Science)
2. (1992). "Academic Press dictionary of science and technology". Gulf Professional Publishing.
3. "algebraic operation {{!}} Encyclopedia.com".
4. William L. Hosch (editor), ''The Britannica Guide to Algebra and Trigonometry'', Britannica Educational Publishing, The Rosen Publishing Group, 2010, {{ISBN
5. James E. Gentle, ''Numerical Linear Algebra for Applications in Statistics'', Publisher: Springer, 1998, {{ISBN. 0387985425, 9780387985428, 221 pages, [James E. Gentle page 183]
6. David Alan Herzog, ''Teach Yourself Visually Algebra'', Publisher John Wiley & Sons, 2008, {{ISBN
7. John C. Peterson, ''Technical Mathematics With Calculus'', Publisher Cengage Learning, 2003, {{ISBN
8. Jerome E. Kaufmann, Karen L. Schwitters, ''Algebra for College Students'', Publisher Cengage Learning, 2010, {{ISBN