Adding Rational Expressions

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An expression which can be represented as a ratio of two polynomials i.e., in numerator there will be a polynomial and in denominator there will be another different polynomial because if the polynomial would be same to both numerator and denominator then the ratio becomes 1 and 1 is not a rational number.


Example of rational expressions: -
·        (2x^2 + 5) / (x + 5)

·        (3x^2 + 2x + 4) / (x^2 + 9x + 8)
·        (x + 2) / (x + 5)
·        1/ (x^2 + 5): - 1 is present to the numerator and 1 is a constant polynomial so this is also a rational expression.
·        (x^3 + 5x^2 – 4x +9): - This expression can be represented as
           (x^3 + 5x^2 – 4x +9) / 1. 1 is a constant polynomial; both numerator and denominator contain polynomial so it is a rational expression.

 
Examples of adding rational expressions: -


Add the following rational expressions: -
 
1)    (x^2 + 1)/(x +1) and 2x/(x +1)
2)    1/x , (3x^2+ 2x + 5)/2x and (2x^2 + 5x + 3)/4x

 
Solution: -
1)    (x^2 + 1) / (x +1) + 2x/ (x +1) =(x^2 + 1 + 2x) / (x + 1)
         = (x^2 + 2x + 1) / (x+ 1)
         = (x + 1)^2 / (x+ 1)
         =  (x+ 1)

2)    1/ x + (3x^2+ 2x + 5) / 2x + (2x^2 + 5x + 3) / 4x
        =[4 + 2((3x^2+ 2x + 5) + (2x^2 + 5x + 3)] / 4x
        = (4 + 6x^2 + 4x + 10 + 2x^2 + 5x + 3) / 4x
        = (8x^2+9x+17)/4x





 

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