EXAMPLES OF ADDING AND SUBTRACTING RATIONAL EXPRESSIONS

Example 1 :

Simplify :

(x + 2)/(x + 3)  +  (x - 1)/(x + 3)

Solution :

Because the denominators are same, we have to take the denominator once and combine the numerators. 

=  (x + 2 + x - 1) / (x + 3)

=  (2x + 1) / (x + 3)

Example 2 :

Simplify :

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

Solution :

Here, the denominators are not same.

So, we have to find the least common multiple of the denominators.

Least common multiple of the denominators is

(x - 1)2(x + 1)

Then, 

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

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

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

=  [x2 + 2x + 1 + x2 - 2x + 1] / (x + 1)(x - 1)2

=  (2x2 + 2) / (x + 1)(x - 1)2

Example 3 :

Simplify

[x3/(x - 2)]  +  [8/(2 - x)]

Solution :

Here, the denominators are not same. But, we can do a small adjustment and  make denominator same as shown below.

  =  (x+ 8)/(x - 2)]  -  8/(x - 2)

Now, the denominators is same. So, we have to take the denominator once and combine the numerators. 

=  [x3/(x - 2)]  -  [8/(x - 2)]

  =  (x3  - 8) / (x - 2)

Use algebraic identity a3 - b3  =  (a - b)(a2 + ab + b2) to factor (x3 - 8). 

  =  (x - 2)(x2 + 2x + 4) / (x - 2)

  =  x2 + 2 x + 4

Example 4 :

Simplify

(x + 2)/(x2 + 3x + 2)]  +  (x - 3)/(x2 - 2x - 3)

Solution :

x2 + 3 x + 2  =  (x + 1)(x +  2)

x2 - 2 x - 3  =  (x - 3)(x + 1)

Then, 


Example 5 :

Simplify

[(x2 - x - 6)/(x2 - 9)] + [(x2 + 2x - 24)/(x2 - x - 12)]

Solution :

  =  [(x2 - x - 6)/(x2 - 9)] + [(x2 + 2x - 24)/(x2 - x - 12)]

(x2 - x - 6)  =  (x - 3)(x + 2)

By comparing (x2 - 9) with the algebraic identity 

(a2 - b2)  =  (a + b)(a - b)

we get,

(x2 - 32)  =  (x + 3)(x - 3)

(x2 + 2 x - 24)  =  (x + 6) (x - 4)

(x2 - x - 12)  =  (x - 4) (x + 3)

Example 6 :

Simplify

[(2x2-5x+3)/(x2-3x+2)] - [(2x2-7x-4)/(2x2 - 3x - 2)]

Solution :

  =  [(2x2-5x+3)/(x2-3x+2)] - [(2x2-7x-4)/(2x2 - 3x - 2)]

Example 7 :

Simplify

[(x2-4)/(x2+6x+8)]-[(x2-11x+30)/(x2-x - 20)]

Solution :

  =  [(x2-4)/(x2+6x+8)] - [(x2-11x+30)/(x2-x - 20)]

(x- 22)  =  (x + 2) (x - 2)

(x2+ 6x + 8)  =  (x + 2) (x + 4)

(x2- 11x + 30)  =  (x - 6) (x - 5)

(x2- x - 20)  =  (x - 5) (x + 4)

Example 8 :

Simplify

[(2x + 5)/(x + 1)] + [(x2 + 1)/(x2 - 1)] - [(3x - 2)/(x - 1)]

Solution :

Example 9 :

Simplify

[1/(x2+3x+2)] + [1/(x2+5x+6)] - [2/(x2+4x+3)]

Solution :

(x+ 3x + 2)  =  (x + 1) (x + 2)

(x+ 5x + 6)  =  (x + 2)(x + 3)

(x+ 4x + 3)  =  (x + 3) (x + 1)

  =  0

Example 10 :

Which rational expression should be added to

(x3 - 1)/(x2 + 2) to get (3x3 + 2x2 + 4)/(x2 + 2) ?

Solution :

let the required rational expression be p(x)

[(x- 1)/(x+ 2)]  + p(x)  =  (3x3 + 2x2 + 4)/(x2 + 2)

p(x)  =  [(3x3 + 2x2 + 4)/(x2 + 2)] - [(x- 1)/(x+ 2)]

Since the denominators are same, we may write only one denominator and combine the numerators.

=  [(3x3 + 2x2 + 4) - (x- 1)]/(x+ 2)]

=  (3x3 - x3 + 2x2 + 4 + 1)/(x2 + 2)

=  (2x3 + 2x2 + 5)/(x2 + 2)

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