# 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.

Rational expressions assingments - 1 part

Rational expressions assingments - 2 part

Formulas for short multiplication

Polynomials & polynomial identities - forums