# Rational Expressions - monomial, polynomial

Different numbers and variables, related with the addition, subtraction, multiplication and division signs are called rational expressions.

For example: $2ab$, $4a^2+34x + z^5$, $\frac{3a+c}{5}$, $\frac{4}{5xy+12}$, $\frac{x+1}{2y+x}$

The first three do not contain the variable in the divisor. They are called whole rational expressions. In the last three examples the variable is in the divisor and they are called fractional rational expression.

Every rational expression in which there is only the multiplication operation (including exponentiation), is called monomial.
Examples: $2ab, a^3, 7x^2y^4, \frac{2x}{y}$.

In case there are the operations of addition and subtraction participating in it – polynomial.
Examples:
$2x + 3$
$2ab + a^3$
$c^2 - 2xa + 93y^2b^5$

Monomials which cannot be represented by a product of smaller number of multipliers, are called normal monomials.
Examples:
$14x^2y^3$ is a normal monomial
$14(xy)xy^2$ is not normal because x and even y are met twice.

Monomials, in which there are one and the same variables to respectively equal powers, are called similar monomials.
Examples:
Similar are $2a^2$ and $10а^2$
$5xy$ and $xy$
$a^3b^2c$ and $20a^2b^2ac$

Polynomials, in which there are no similar monomials, are called normal polynomials.
Example: $5x^2 + 7x$
$60xy^2 + 34x - 10$

If, in a given rational expression, the unknowns are substituted for definite numerical values and the actions marked are performed, then a number is received called numerical value of the expression.
Example:
If $x = 2$ and $y = 3$ the numerical value of the expression $x^2y + 2y^2$ is $x^2y + 2y^2 = 2^2\cdot 3 + 2\cdot 3^2 = 12 + 18 = 30$.

Two rational expressions are equivalent, if their respective values are equal for all the values of the variables participating in them.
Example:
$(x-y)^2 = x^2 -2xy-y^2$
So the expressions $(x-y)^2$ and $x^2 -2xy-y^2$ are equivalent.

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