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A rational expression is defined as the ratio of two polynomials. For example , here is a polynomial of degree 2 and x + 1 is a polynomial of degree 1. For polynomials, the exponents of variables are always positive. is not a rational expression as is not a polynomial.

**Example 1:** Multiply the rational expressions and .

Multiplication of rational expressions:

X

Factoring the expressions. Factors of are (x + 3) (x – 3)

This is of the form - = (a + b) ( a – b). If we compare - with - then we get a = x and b = 3. Hence the factors are (x + 3) (x – 3)

Factors of are = (x + 1) (x + 1)

X

= (x – 3) (x + 1)

= + x – 3x – 3

= - 2x – 3

Hence the product of given rational expression is - 2x – 3.

**Example 2:** Dividing the rational expressions and.

Division of rational expressions: ÷

=

= X

Writing the expressions into factors.

= (x + 5) (x + 2)

= (x + 7) (x – 2)

= X

=

= (x+ 5) (x + 7)

Multiplication using FOIL method,

Multiplication of first terms x and x =

Multiplication of outer terms x and 7 = 7x

Multiplication of Inner terms 5 and x = 5x

Multiplication of Last terms 5 and 7 = 35

Adding all the terms = + 7x + 5x + 35 = + 12x + 35