# SIMPLIFYING RATIONAL EXPRESSIONS

Simplifying rational expressions :

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

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

## Simplifying rational expressions - Steps

Step 1 :

Factor both numerator and denominator, if possible.

Step 2 :

Identify the common factor at both numerator and denominator.

Step 3 :

The common factor identified at both numerator and denominator should be multiplied by the other terms.

Step 4 :

Now, get rid of the common factor at both numerator and denominator.

The above four steps have been illustrated in the picture given below. ## Simplifying rational expressions - Examples

Example 1 :

Write the simplified expression for the function defined by

f(x)  =   4x³ / 8x²

Solution:

In this example we have  x³  and x² . We have to split these two terms as much as possible.

f(x)  =  4.x.x.x / 8 x.x

Getting rid of two x's at both numerator and denominator, and simplifying 4/8, we we get

f(x)  =  x / 2

Example 2 :

Write the simplified expression for the function defined by

f(x)  =   (5x + 20) / (7x + 28)

Solution:

Factoring the numerator and denominator of the given rational function, we get

f(x)  =   5(x + 4) / 7(x + 4)

Getting rid of the common factor (x + 4) at both numerator and denominator, we get

f(x)  =  5 / 7

Example 3 :

Write the simplified expression for the function defined by

f(x)  =   (3x + 9) / (3x + 15)

Solution:

Factoring the numerator and denominator of the given rational function, we get

f(x)  =   3(x + 3) / 3(x + 5)

Getting rid of the common factor 3 at both numerator and denominator, we get

f(x)  =  (x + 3) / (x + 5)

Example 4 :

Write the simplified expression for the function defined by

f(x)  =   (9x² - 25y²) / (3x² - 5xy)

Solution:

f(x)  =   (9x² - 25y²) / (3x² - 5xy)

f(x)  =    [(3x)² - (5y)²] / (3x² - 5xy)

Factoring the numerator and denominator of the given rational function, we get

f(x)  =    [(3x + 5y)(3x - 5y)] / x(3x - 5y)

Getting rid of the common factor (3x - 5y) at both numerator and denominator, we get

f(x)  =  (3x + 5y) / x

Example 5 :

Write the simplified expression for the function defined by

f(x)  =   (6x² - 54) / (x² + 7x + 12)

Solution:

Factoring the numerator and denominator of the given rational function, we get

f(x)  =   6(x² - 9) / (x + 3)(x + 4)

f(x)  =   6(x² - 3²) / (x + 3)(x + 4)

f(x)  =   6(x + 3)(x - 3) / (x + 3)(x + 4)

Getting rid of the common factor (x + 3) at both numerator and denominator, we get

f(x)  =  6(x-3) / (x + 4)

Example 6 :

Write the simplified expression for the function defined by

f(x)  =   (64a³ + 125b³) / (4a²b + 5ab²

Solution:

f(x)  =   (64a³ + 125b³) / (4a²b + 5ab²

f(x)  =   (4³a³ + 5³b³) / (4a²b + 5ab²)

f(x)  =   [(4a)³ + (5b)³] / (4a²b + 5ab²)

Factoring the numerator and denominator of the given rational function, we get

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

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

f(x)  =   [16a² - 20ab + 25b²] / ab

Getting rid of the common factor (4a + 5b) at both numerator and denominator, we get

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

f(x)  =   [16a² - 20ab + 25b²] / ab

Example 7 :

Write the simplified expression for the function defined by

f(x)  =   (x² + 7x + 10) / (x² -  4

Solution:

f(x)  =   (x² + 7x + 10) / (x² -  4)

f(x)  =   (x² + 7x + 10) / (x² - 2²)

Factoring the numerator and denominator of the given rational function, we get

f(x)  =   (x + 2)(x + 5) / (x + 2)(x - 2)

Getting rid of the common factor (x + 5) at both numerator and denominator, we get

f(x)  =   (x + 5) / (x - 2)

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