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Rational Functions

Equivalent Rational Functions

We assume that you are familiar with the arithmetic operations of fractions of numbers.

A rational function is the ratio of two polynomials.

Examples

(x+1)/(x-1) ,     1/((x+1)(x+2)) ,    (x^2+2*x-3)/(x^2-9) ,     x^2+1=(x^2+1)/1.

The top part of a rational function is called the numerator and the bottom part the denominator.

Reduction

As with fractional numbers, we can simplify (reduce) a rational function by cancelling common factors found in both the numerator and denominator of the function (dividing by a common factor):

(a*c)/(b*c)=a/b

Example

(12*(x^2+1)*(x+3))/(15*x^2*(x+3))=(4*(x^2+1))/(5*x^2)

More Examples (this opens in another window)

Exercise

Simplify the rational function below.  Use computer notation.
   
(x2+ )
  
( x+ )
  
( x+ )
  
=
 
(x2+ )
  
( x+ )
  
 ( x+ )
  

 

The technique of factorization of a polynomial is often needed.

Examples

(30*x^2-18*x)/(12*x^2)=(6*x*(5*x-3))/(12*x^2)=(5*x-3)/(2*x).
(x^2+2*x-3)/(x^2-9)=((x+3)*(x-1))/((x+3)*(x-3))=(x-1)/(x-3).
(x^2-x-6)/(x^2-4*x+3)=((x+2)*(x-3))/((x-1)*(x-3))=(x+2)/(x-1).
(4*x^2-5*x-6)/(x^2-4*x+4)=((4*x+3)*(x-2))/((x-2)^2)=(4*x+3)/(x-2).

Exercise

Simplify the rational function below.

x2  x    
=
x2  x    

stop. A common mistake: Cancelling part of a sum or difference.  ONLY common factors can be cancelled.

(3*x^2+2)/x=3*x+2 Wrong(3*x^2)/x=(3*x^2)/x + 2/x = 3*x + 2/x Right

 

 

Rational Functions Index | Addition and Subtraction of Rational Fractions >>

 
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