ADDITION AND SUBTRACTION OF RATIONAL EXPRESSIONS EXAMPLES

Adding and Subtracting Rational Expressions with Like Denominators :

(i) Add or Subtract the numerators

(ii) Write the sum or difference of the numerators found in step (i) over the common denominator.

(iii) Reduce the resulting rational expression into its lowest form

Adding and Subtracting Rational Expressions with unlike Denominators :

(i) Determine the Least Common Multiple of the denominator.

(ii) Rewrite each fraction as an equivalent fraction with the LCM obtained in step (i). This is done by multiplying both the numerators and denominator of each expression by any factors needed to obtain the LCM.

(iii) Follow the same steps given for doing addition or subtraction of the rational expression with like denominators.

Example 1 :

Which rational expression should be subtracted from

(x2 + 6x + 8)/(x3 + 8) to get 3/(x2 - 2x + 4)

Solution :

Let p(x) be the rational expression should be subtracted.

Example 2 :

If A  =  (2x + 1)/(2x - 1) and B  =  (2x - 1)/(2x + 1) find 1/(A - B) - 2B/(A2 - B2)

Solution :

=  1/(A - B) - 2B/(A2 - B2)

=  [1/(A - B)] - [2B/(A + B)(A - B)]

=  (A + B) - 2B/(A + B) (A - B)

=  (A + B - 2B) / (A + B) (A - B)

=  (A - B) / (A + B) (A - B)

=  1/(A + B)

Now let us apply the values of A and B.

  =  1/[(2x + 1)/(2x - 1) + (2x - 1)/(2x + 1)]

  =   1/[(2x+ 1)2 + (2x - 1)2/(2x + 1)(2x - 1)]

  =   (2x + 1)(2x - 1)/[(2x+ 1)2 + (2x - 1)2]

  =  (4x2 - 1)/[4x2 + 4x + 1 + 4x2 - 4x + 1]

  =  (4x2 - 1)/(8x+ 2)

  =  (4x2 - 1)/2(4x+ 1)

Example 3 :

If A = x/(x + 1), B = 1/(x + 1), prove that

((A + B)2 + (A- B)2)/(A ÷ B)  =  2(x2 + 1)/x(x + 1)2

Solution :

(A + B)2 + (A- B) =  (A2 + 2AB + B2) + (A2 - 2AB + B2)

=   A2 + 2AB + B2 + A2 - 2AB + B2

=  2(A2 + B2)

((A + B)2 + (A- B)2)/(A ÷ B)  =  2(A2 + B2)/(A ÷ B) 

A2 + B2 = (x/(x + 1))2 + (1/(x + 1))2

  =  (x/(x + 1)2) + (1/(x + 1)2)

  =  (x2 + 1)/(x + 1)2

2(A2 + B2)  =  2(x2 + 1)/(x + 1)2  ------(1)

(A ÷ B)  =  [x/(x + 1)] ÷ [1/(x + 1)]

  =  x/(x + 1) ⋅ (x + 1)/1

  =  x  ------(2)

(1)/(2)

  =   2(x2 + 1)/(x + 1)2/x

  =   2(x2 + 1)/x(x + 1)2

Hence proved.

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