# ADDING AND SUBTRACTING RATIONAL EXPRESSIONS

The following steps will be useful to add and subtract rational expressions.

Step 1 :

Check whether the denominators of two rational expressions are same.

• If they are same, then put only one common denominator and combine the numerators.
• If the denominators are not same, then we need to take least common multiple. For that we need to factor them into linear factor.

Step 2 :

After taking least common multiple the denominator will become same. So, put only one denominator and combine the numerators.

## Adding Rational Expressions with Like Denominators

Example :

Simplify :

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

Solution : Because the denominators are same, we take the denominator once and combine the numerators.

## Adding Rational Expressions with Unlike Denominators

Example :

Simplify :

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

Solution : To make the denominators same, we need to take least common multiple. ## Practice Questions

Question 1 :

Simplify :

[x3 / (x - 2)] + [8 / (2-x)]

=  [x3 (x - 2)] + [8 / (2-x)]

=  [x/ (x - 2)] - [8 / (x-2)]

=  [ (x³  - 8)/(x - 2) ]

By comparing x3  - 8 with the algebraic identity

a3 - b3  =  (a - b) (a2 + ab + b2)

we get,

=  (x - 2) (x2 + x(2) + 22) / (x - 2)

=  (x2 + 2 x + 4)

Question 2 :

Simplify :

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

Solution :

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

(x² + 3 x + 2)  =  (x + 1) (x +  2)

(x² - 2 x - 3)  =  (x - 3) (x + 1) Question 3 :

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) Question 4 :

Simplify

[(x - 2)/(x2 - 7x + 10)] + [(x + 3)/(x2 - 2x - 15)]

Solution :

=  [(x - 2)/(x2 - 7x + 10)] + [(x + 3)/(x2 - 2x - 15)]

(x² - 7 x + 10)  =  (x - 2 ) (x - 5)

(x² - 2 x - 15)  =  (x - 5) (x + 3) Question 5 :

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)] Question 6 :

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) Question 7 :

Simplify

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

Solution :  Question 8 :

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

Question 9 :

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)

Question 10 :

Which rational expression should be subtracted from

(4x3 - 7 x2 + 5)/(2x - 1) to get 2x2 - 5x + 1 ?

Solution :

Let p(x) be the required rational expression

(4x3 - 7 x2 + 5)/(2x - 1) - p(x)  =  2x2 - 5x + 1

(4x3 - 7 x2 + 5)/(2x - 1) - (2x2 - 5x + 1)  =  p(x)

p(x)  =  [(4x3 - 7 x2 + 5) - (2x2 - 5x + 1)(2x - 1)]/(2x - 1) Question 11 :

If P  =  x/(x + y) Q  =  y/(x + y), then find

[1/(P-Q)] - [2Q/(P²-Q²)]

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