ADDING AND SUBTRACTING RATIONAL EXPRESSIONS

Adding and subtracting rational expressions :

In this section, we will learn how to add and subtract rational expressions.

Steps Involved in Adding and Subtracting 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

Question 1 :

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

Solution :

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

Adding rational expressions with unlike denominators

Question 2 :

Simplify [(x+1)/(x-1)²] + [1/(x+1)]

Solution :

To make the denominators same, we need to take least common multiple.

More examples of adding rational expressions with different denominators

Question 3 :

Simplify

[x³ / (x - 2)] + [8 / (2-x)]

Solution :

  =  [x³ / (x - 2)] + [8 / (2-x)]

  =  [x³ / (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 4 :

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 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)

Question 6 :

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)

Example of Subtracting Rational Expressions

Question 7 :

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 8 :

Simplify

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

Solution :

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

(x² - 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 9 :

Simplify

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

Solution :

Question 10 :

Simplify

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

Solution :

(x² + 3x + 2)  =  (x + 1) (x + 2)

(x² + 5x + 6)  =  (x + 2)(x + 3)

(x² + 4x + 3)  =  (x + 3) (x + 1)

  =  0

Question 11 :

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)

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

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

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

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

=  (3x³ - x³ + 2x² + 4 + 1)/(x² + 2)

=  (2x³ + 2x² + 5)/(x² + 2)

Question 12 :

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 13 :

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

[1/(P-Q)] - [2Q/(P²-Q²)]

Solution :

After having gone through the stuff given above, we hope that the students would have understood how to add and subtract rational expressions.

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