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 to lowest term or simplest form.

The following steps ill be useful to simple rational expressions. 

Step 1 :

Factor both numerator and denominator, if possible.

Step 2 :

Identify the common factors in both numerator and denominator. 

Step 3 :

Remove the common factors found in both numerator and denominator.

The above four steps have been illustrated in the picture given below. 

Practice Questions

Question 1 : 

Simplify :

4x³ / 8x²

Answer : 

=  4x³ / 8x²

=  (1/2)x^3-2

=  (1/2)x

=  x/2

Question 2 : 

Simplify :

(5x + 20) / (7x + 28) 

Answer : 

=  (5x + 20) / (7x + 28)

=  5(x + 4) / 7(x + 4) 

=  5/7

Question 3 : 

Simplify :

(3x + 9) / (3x + 15) 

Answer : 

=  (3x + 9) / (3x + 15) 

=  3(x + 3) / 3( x + 5)

=  (x + 3) / (x + 5)

Question 4 : 

Simplify : 

(9x² - 25y²) / (3x² - 5xy)

Answer : 

=  (9x² - 25y²) / (3x² - 5xy)

=  [(3x)² - (5y)²] / x(3x - 5y)

=  [(3x + 5y)(3x - 5y)] / x(3x - 5y)

=  (3x + 5y) / x

Question 5 : 

Simplify :

(6x² - 54) / (x² + 7x + 12) 

Answer : 

=  (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)

Question 6 : 

Simplify : 

(64a³ + 125b³) / (4a²b + 5ab²)

Answer : 

=  (64a³ + 125b³) / (4a²b + 5ab²)

=  [(4a)³ + (5b)³] / ab(4a + 5b)

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

=  (16a² - 20ab + 25b²) / ab

Question 7 : 

Simplify :

(x² + 7x + 10) / (x² -  4) 

Answer : 

=   (x² + 7x + 10) / (x² -  4)

=  (x + 5)(x + 2) / (x² - 2²)

=  (x + 5)(x + 2) / (x + 2)(x - 2)

=   (x + 5) / (x - 2)

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