Algebraic Expressions
Algebraic Expressions
In mathematics, an algebraic expression is an expression built op from constants, variables, and a finite
number of algebraic operations (addition, subtraction, multiplication, division and exponentiation to a
power that is a rational number).
A rational algebraic expression (or rational expression) is an algebraic expression that can be written as
a quotient of polynomials, such as x2 + 2x + 4. An irrational algebraic expression is one that is not
rational, such as x + 4.Some but not all polynomial equations with rational coefficients have a solution
that is an algebraic expression with a finite number of operations involving just those coefficients (that
is, can be solved algebraically). This can be done for all such equations of degree one, two, three, or
four; but for degree five or more it can only be done for some equations but not for all.
Analytical expression:-In mathematics, an analytical expression (or expression in analytical form) is a
mathematical expression constructed using well-known operations that lend themselves readily to
calculation. As is true for closed-form expressions, the set of well-known functions allowed can vary
according to context but always includes the basic arithmetic operations (addition, subtraction,
multiplication, and division), extraction of nth roots, exponentiation, logarithms, and trigonometric
functions.
Know More About :- Whole Numbers


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However, the class of expressions considered to be analytical expressions tends to be wider than that for
closed-form expressions. In particular, special functions such as the Bessel functions and the gamma
function are usually allowed, and often so are infinite series and continued fractions. On the other hand,
limits in general, and integrals in particular, are typically excluded.
If an analytic expression involves only the algebraic operations, which are addition, subtraction,
multiplication, division and exponentiation with integral or fractional exponents (hence including the
extraction of nth roots), then it is more specifically referred to as an algebraic expression.
In the case of one variable, , a function is called a rational function if and only if it can be written in the
form where and are polynomial functions in and is not the zero polynomial. The domain of is the set
of all points for which the denominator is not zero, where one assumes that the fraction is written in its
lower degree terms, that is, and have several factors of the positive degree.
Every polynomial function is a rational function with . A function that cannot be written in this form
(for example, ) is not a rational function (but the adjective "irrational" is not generally used for
functions, but only for numbers).An expression of the form is called a rational expression. The need
not be a variable. In abstract algebra the is called an indeterminate.
A rational equation is an equation in which two rational expressions 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.
A constant function such as f(x) = is a rational function since constants are polynomials. Note that the
function itself is rational, even though f(x) is irrational for all x.The rational function is equal to 1 for
all x except 0, where there is a removable singularity.The sum, product, or quotient (excepting division
by the zero polynomial) of two rational functions is itself a rational function: however, the process of
reduction to standard form may inadvertently result in the removing of such discontinuities unless care
is taken.
Read More About :- Trigonometric Functions


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