Rational Algebraic Expressions

A rational expression is an algebraic expression of the `(P)/(Q)` from ,where P and Q are simpler expressions (usually polynomials), and the denominator Q is not zero.



This rational expressions whose numerators and denominators are (or written as) polynomials. Like polynomials, rational expressions appear frequently in the Algebra II and higher mathematics. We must understand the how to perform basic operations with rational expressions and how to solve rational equations.


Two Types:-


1.Adding and Subtracting Rational Expressions

2.Multiplying and Dividing Rational Expressions



1. Adding and Subtracting Rational Expressions

Adding Subtracting Rational Expressions with the Like Denominators

Add and subtract their numerators and then write the results over the denominator. Then, simplify the numerator factor, and write the expression in lowest terms.

Adding and Subtracting Rational Expressions with the Unlike Denominators

Rational expressions can be added and subtracted if and only if they have the same denominator. Thus, we must rewrite them as expressions with a common denominator.



2.Multiplying and Dividing Rational Expressions

Multiplying Rational Expressions

To multiply two rational expressions, factor them. Then multiply their numerators and denominators, crossing out the any factors that appear in both the numerator and the denominator

Dividing Rational Expressions

To divide by rational expression,and multiply by its reciprocal.


Examples:-


Example 1:

Simplify the rational expression

4x - 2
--------
2x - 1
Solution :-

Factor both the numerator and denominator completely.

2(2x - 1)
---------
2x - 1
Cancel common factors to reduce and then simplify the given expression.

2(2x - 1)
= ------------ = 2 , with x not equal to 1
2x - 1


Example 2:

Simplify the rational expression

4x + 16
---------
x2 - 16
Solution:-

Factor both the numerator and denominator completely.

4x + 16         4(x + 4)
--------- = ----------------
x2  - 16      (x + 4)(x - 4)
Cancel common factors to simplify the expression.

4(x + 4)
= ---------------
(x - 4)(x + 4)


4
= --------    , with x not equal to -4.
(x - 4)

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Example 3:

Simplify the rational expression

x2 + x - 2
-------------
-x2 -2x + 3

Solution:-

Factor both the numerator and denominator completely.

x2 + x - 2      (x + 2)(x - 1)
------------- = ----------------
-x2 -2x + 3      (x + 3)(-x + 1)


Note that - x + 1 = - (x - 1) in the denominator.

(x + 2)(x - 1)
= -----------------
-(x + 3)(x - 1)


Cancel common factors.

(x + 2)(x - 1)
=--------------------
-(x + 3)(x - 1)


(x + 2)
= -------------   , with x not equal to 1
(x + 3)