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.

**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.

**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.

**Question 2 :**

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

**Solution :**

To make the denominator same, we need to take L.C.M

**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)

**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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