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)]

Answer :

  =  [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 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)]

(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 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 :

(x² + 3x + 2)  =  (x + 1) (x + 2)

(x² + 5x + 6)  =  (x + 2)(x + 3)

(x² + 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)

[(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 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)]

Solution :

Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

You can also visit the following web pages on different stuff in math. 

WORD PROBLEMS

HCF and LCM  word problems

Word problems on simple equations 

Word problems on linear equations 

Word problems on quadratic equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation 

Word problems on unit price

Word problems on unit rate 

Word problems on comparing rates

Converting customary units word problems 

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles 

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems 

Profit and loss word problems 

Markup and markdown word problems 

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed 

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS 

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6