# Simplifying Rational Expressions

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Simplifying Rational Expressions
In this page we are going to discuss about simplifying rational expressions concept . A rational expression
is an expression of the form P(x) / Q(x) over the set of real numbers and Q(x) ? 0 where P(x) and Q(x) are two
polynomials. It is the quotient of two polynomials.
For example: 3 / x2 ,
(x4 + x3 + x + 1) / (x + 6) ,
5 / (x + 8) ,
(x - 2) / (x + 2) ,
28y-3 are all rational algebraic expressions.
How to simplify rational expressions
Let's see some examples on how to simplify rational expressions -
Rational algebraic expressions P(x) / Q(x) can be reduced to its lowest term by dividing both numerator P(x)
and denominator Q(x) by the G.C.D. of P(x) and Q(x).
Example 1:
Simplify the rational algebraic expression (5x + 20) / (8x + 32)
Solution:
(5x + 20) / (8x + 32) = 5(x + 4) / 8(x + 4)
Cancel (x + 4) term from both numerator and denominator,
(5x + 20) / (8x + 32) = 5 / 8
Example 2:
Simplify (5x + 20) / (6x + 24)
Solution:
(5x
+ 20) / (6x + 24) = 5(x + 4) / 6(x + 4)

Cancel (x + 4) term from both numerator and denominator,
(5x + 20) / (6x + 24) = 5 / 6

Example 3:
Simplify the expression (4x + 16) / (4x + 24)
Solution:
(4x + 16) / (4x + 24) = 4(x + 4) / 4(x + 6)
Cancel the number 4 from both numerator and denominator, simplifying rational expressions
(4x + 16) / (4x + 24) = (x + 4) / (x + 6)
Example 4:
Simplify rational algebraic expression (x2 - x - 6) / (x2 + 5x + 6)
Solution:
(x2 - x - 6) / (x2 + 5x + 6) = (x - 3)(x + 2) / (x + 2)(x + 3)
Cancel (x + 2) term from both numerator and denominator,
(x2 - x - 6) / (x2 + 5x + 6) = (x -3) / (x + 3)
Example 5:
Simplify (x2 + 7x + 10) / (x2 - 4)
Solution:
(x2 + 7x + 10) / (x2 - 4) = (x + 2)(x +5) / (x2 - 22)
Term (x2 - 22) can be written as (x + 2)(x - 2),
= (x + 2)(x +5) / (x + 2)(x - 2)
Cancel (x + 2) term from both numerator and denominator,
= (x + 5) / (x - 2)
Practice Problems
Simplifying the following rational algebraic expressions,
1) (2x + 10) / (2x - 6)
2) (6x + 24) / (9x + 36)
3) (3a4b) / (a3b2)
4) (x4 - x2y2) / (y4 - x2y2)
Solutions:
1) (x + 5) / (x - 3)
2) 6 / 9
3) 3a / b
4) - x2 / y2

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