Rational expression

 Rational expression in the form of p/q. Rational expression show the ratio of two polynomials . It means both the numerator and denominator are polynomials in it. It is also ratio of algebraic expression , which consist of unknown variables . for example( x+3)/6  , ( x² +x+6)/6x  etc. 

To find the root or zero polynomial expressions , We have to put them equal to zero .  But to find the zeros of rational functions or expressions, we have to put only the numerator equal to zero when the expression has been reduced to its lowest terms.

By lowest terms, we mean that both the numerator and denominator do not have any common factors. Like in the case of a fraction, say 4/16, it is not in the lowest form. It can be further reduced by taking 4 as a common factor. Thus, ¼ is the lowest form.

Simplification of Rational Expression 

 Rational expression performe the basic arithmetic operations such as, addition , subtraction , multiplication and division. 

Addition and subtraction : Rational expression having the same denominator , keep the denominator as common and add or subtract the numerator . And find their lowest term if possible .

If the rational expression have different denominator , then first find LCM. Now , change each rational expression to the equivalent one by making the denominator exactly the same . Finally , add or subtract the terms and their lowest term if possible .

Generally , we express the addition and subtraction by the given formula :

  • a/c + b/c = (a+ b) / c and a/c - b/c = (a -b)/ c  
If same denominator 

ex..,   for addition 
            (x² - 4 / x + 2 ) + (x² - 9 / x +2 )
 = {(x +2)(x - 2)/ (x +2) }+ {(x + 3)(x - 3)/(x +2)}
  = {(x+2) (x - 2) + ( x + 3)(x -3)}/(x + 2) 
  = (x - 2) + (x + 3)(x - 3)
  = x - 2 + x² - 9
  = x² + x -11

for subtraction 

ex.,      (x² - 4 / x + 2 ) - (x² - 9 / x +2 )

         = {(x +2)(x - 2)/ x +2} - {(x + 3)(x - 3)/x +2}
         = {(x +2) (x - 2) - ( x + 3)(x -3)} /                   (x + 2) 
         = (x - 2) - (x + 3)(x - 3)
         = x - 2 - x² + 9 
         = x² - x +7

For different denominator  

For addition 

    (x² − 25/x + 5) + (x² − 36/x - 6)

    ={ (x +5 ) (x- 5)/ x+5 }  + {(x +6)(x                 -6)/x- 6}

    = (x - 5) +(x+ 6) 

    = 2x + 1 

For subtraction 

Ex..,    (x² − 25/x + 5) - (x² − 36/x - 6) 

= (x +5 ) (x- 5)/ x+5 } - {(x +6)(x -6)/x-         6}

= (x - 5) - (x+ 6) 

 = -11

Multiplication :  In multiplication first find the factor of both polynomial (if exist); then reduce the same factor of polynomials in numerator and denominator . Multiplying the remaining numerator and denominator seperately together will result in reduced form .

  • a/b × c/d = ac/bd

Ex.,     

 (x² − 25/x + 5) × (x² − 36/x - 6)

={ (x +5 ) (x- 5)/ x+5 } × {(x +6)(x -6)/x- 6} 

= (x - 5) ×(x+ 6) 

= x²+x- 30

Division : In division first get the inverse of denominator i.e. divisor and multiply with the numerator i.e. dividend because reducing is easily only after converting the division into multiplication , similar to the case of dividing factor . Further simplification is similar to multiplication , as explained above .

  • a/b ÷ c/d = a/b × d/c = ad/bc

Ex.., 

    (x² − 25/x + 5) ÷ (x² − 36/x - 6)


={(x +5 ) (x- 5)/ x+5 )} ×{(x -6)/(x+ 6)(x- 6) }

= (x -5) ×{1/(x + 6)}

= x -5 / x +6 




 








 

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