ADDITION AND SUBTRACTION OF RATIONAL EXPRESSIONS EXAMPLES

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)/(8x² + 2)

  =  (4x² - 1)/2(4x² + 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)²  =  (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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