1. Rational Functions and their Manipulations

2. Q are polynomials and Q  0, is called a rational function.
Rational Functions
A function that can be
written as ,where P and
Q are polynomials and Q  0, is called a rational function. Q P
Example: 2 3 – x
(i)
 3 and x – 2 are polynomials. 5 3 2 – + x y
(ii)
 x2 – y and x3 + x – 5 are polynomials.

3. Do you remember the techniques in simplifying a rational function?
 Factorize both the numerator
and the denominator first.
 Cancel out the common factor.

4. Multiplication and Division of Rational Functions
Before doing multiplication and division, reduce each rational function to its simplest form first. 2
 Simplify each rational function.
 Cancel out the common factors.

5. Follow-up question Simplify .  Simplify each rational function.
 Cancel out the common factors.

6. Addition and Subtraction of Rational Functions
The steps for addition and subtraction of rational functions are shown in the following example.
Step 1: Simplify each rational function.
Step 2: Find the L.C.M. of the denominators.
Step 3: Change each rational function using the L.C.M. as its denominator. ) 7 )( 3 2 ( - + = x

7. Addition and Subtraction of Rational Functions
Step 4: Find the algebraic sum of the new numerators. (For subtraction, find the algebraic difference.)
Step 5: Simplify the result.

8. Follow-up question Simplify .  Simplify each rational function.
 L.C.M. of (x + y)(2x – y) and (x + y)
= (x + y)(2x – y)