Rational Functions and their Manipulations

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.

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

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.

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

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

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.

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