# SIMPLIFYING RATIONAL FUNCTIONS

The function which is in the form of f(x) / g(x) is called rational function.

Simplifying rational function is nothing but expressing the the rational function to its lowest term or in its simplest form.

The following steps would be useful to simplify rational functions.

Step 1 :

Factor both numerator and denominator, if possible.

Step 2 :

Identify the common factor in both numerator and denominator.

Step 3 :

Cancel out the common factors found in numerator and denominator.

The above steps have been illustrated in the picture below.

## Solved Examples

Example 1 :

Simplify :

4x3 / 8x2

Solution :

=  4x3 / 8x2

=  x/2

Example 2 :

Simplify :

(5x + 20) / (7x + 28)

Solution :

=  (5x + 20) / (7x + 28)

=  5(x + 4) / 7(x + 4)

=  5/7

Example 3 :

Simplify :

(3x + 9) / (3x + 15)

Solution :

=  (3x + 9) / (3x + 15)

=  3(x + 3) / 3(x + 5)

=  (x + 3) / (x + 5)

Example 4 :

Simplify :

(9x2 - 25y2) / (3x2 - 5xy)

Solution:

=  (9x2 - 25y2) / (3x2 - 5xy)

=  [(3x)2 - (5y)2] / (3x2 - 5xy)

=  [(3x + 5y)(3x - 5y)] / x(3x - 5y)

=  (3x + 5y) / x

Example 5 :

Simplify :

(6x2 - 54) / (x2 + 7x + 12)

Solution:

=  (6x2 - 54) / (x2 + 7x + 12)

=  6(x2 - 9) / (x + 3)(x + 4)

=  6(x2 - 32) / (x + 3)(x + 4)

=  6(x + 3)(x - 3) / (x + 3)(x + 4)

=  6(x - 3) / (x + 4)

Example 6 :

Simplify :

(64a3 + 125b3) / (4a2b + 5ab2

Solution:

=  (64a3 + 125b3) / (4a2b + 5ab2

=  [(4a)3 + (5b)3] / (4a2b + 5ab2

=  (4a + 5b)[(4a)2 - (4a)(5b) + (5b)2] / ab(4a + 5b)

=  [(4a)2 - (4a)(5b) + (5b)2] / ab

=  (16a2 - 20ab + 25b2 / ab

Example 7 :

Simplify :

=  (x2 + 7x + 10) / (x2 -  4

Solution:

=  (x2 + 7x + 10) / (x2 -  4)

=  (x2 + 7x + 10) / (x2 - 22)

=  (x + 2)(x + 5) / (x + 2)(x - 2)

=  (x + 5) / (x - 2)

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