# ADDING AND SUBTRACTING RATIONAL EXPRESSIONS

## About "Adding and subtracting rational expressions"

On the webpage "adding and subtracting rational expressions" we are going to see some example problem to understand how to add or subtraction rational expressions.

## Steps involved in adding and subtracting rational expressions

Step 1 :

Check whether the denominators of two rational expressions are same.

If it is same, then put only one common denominator and combine the numerators.

Step 2 :

If the denominators are not same, then we need to take L.C.M for that we need to factor them into linear factor.

Step 3 :

After taking L.C.M 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 denominator same, we need to take L.C.M ## 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) ]

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)

Let us see the next example problem on "Adding and subtracting rational expressions".

Question 4 :

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

Solution :

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

(x² + 3 x + 2) = (x + 1) (x +  2)

(x² - 2 x - 3) = (x - 3) (x + 1) Let us see the next example problem on "Adding and subtracting rational expressions".

Question 5 :

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

Solution :

= [(x² - x - 6)/(x² - 9)] + [(x² + 2 x - 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) Let us see the next example problem on "Adding and subtracting rational expressions".

Question 6 :

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

Solution :

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

Solution :

= [(2x²-5x+3)/(x²-3x+2)]-[(2x²-7x-4)/(2x² - 3 x - 2)] Let us see the next example problem on "Adding and subtracting rational expressions".

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+ 6 x + 8) = (x + 2) (x + 4)

(x²-11x+30) = (x - 6) (x - 5)

(x²-x - 20) = (x - 5) (x + 4) Let us see the next example problem on "Adding and subtracting rational expressions".

Question 9 :

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

Solution :  Let us see the next example problem on "Adding and subtracting rational expressions".

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

Let us see the next example problem on "Adding and subtracting rational expressions".

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)]

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

=  (3x³ - x³ + 2x² + 4 + 1)/(x² + 2)

=  (2 x³ + 2 x² + 5)/(x² + 2)

Let us see the next example problem on "Adding and subtracting rational expressions".

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) Let us see the next example problem on "Adding and subtracting rational expressions".

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 "Adding and subtracting rational expressions".

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