3 Addition and subtraction of algebraic fractions

To add two algebraic fractions the lowest common denominator must be found first. This is the simplest algebraic expression that has the given denominators as its factors. All fractions must be written with this lowest common denominator. Their sum is found by adding the numerators and dividing the result by the lowest common denominator.

To subtract two fractions the process is similar. The fractions are written with the lowest common denominator. The difference is found by subtracting the numerators and dividing the result by the lowest common denominator.

Example 57

State the simplest expression which has x + 1 and x + 4 as its factors.

Solution

The simplest expression is ( x + 1 ) ( x + 4 ) . Note that both x + 1 and x + 4 are factors.

Example 58

State the simplest expression which has x − 1 and ( x − 1 ) 2 as its factors.

Solution

The simplest expression is ( x − 1 ) 2 . Clearly ( x − 1 ) 2 must be a factor of this expression. Also, because we can write ( x − 1 ) 2 = ( x − 1 ) ( x − 1 ) it follows that x − 1 is a factor too.

Example 59

Express as a single fraction 3 x + 1 + 2 x + 4

Solution

The simplest expression which has both denominators as its factors is ( x + 1 ) ( x + 4 ) . This is the lowest common denominator. Both fractions must be written using this denominator. Note that 3 x + 1 is equivalent to 3 ( x + 4 ) ( x + 1 ) ( x + 4 ) and 2 x + 4 is equivalent to 2 ( x + 1 ) ( x + 1 ) ( x + 4 ) . Thus writing both fractions with the same denominator we have

3 x + 1 + 2 x + 4 = 3 ( x + 4 ) ( x + 1 ) ( x + 4 ) + 2 ( x + 1 ) ( x + 1 ) ( x + 4 )

The sum is found by adding the numerators and dividing the result by the lowest common denominator.

3 ( x + 4 ) ( x + 1 ) ( x + 4 ) + 2 ( x + 1 ) ( x + 1 ) ( x + 4 ) = 3 ( x + 4 ) + 2 ( x + 1 ) ( x + 1 ) ( x + 4 ) = 5 x + 14 ( x + 1 ) ( x + 4 )

Key Point 21
Addition of two algebraic fractions

Step 1: Find the lowest common denominator

Step 2: Express each fraction with this denominator

Step 3: Add the numerators and divide the result by the lowest common denominator

Example 60

Express 1 x − 1 + 5 ( x − 1 ) 2 as a single fraction.

Solution

The simplest expression having both denominators as its factors is ( x − 1 ) 2 . We write both fractions with this denominator.

1 x − 1 + 5 ( x − 1 ) 2 = x − 1 ( x − 1 ) 2 + 5 ( x − 1 ) 2 = x − 1 + 5 ( x − 1 ) 2 = x + 4 ( x − 1 ) 2

Task!

Express 3 x + 7 + 5 x + 2 as a single fraction.

First find the lowest common denominator:

( x + 7 ) ( x + 2 )

Re-write both fractions using this lowest common denominator:

3 ( x + 2 ) ( x + 7 ) ( x + 2 ) + 5 ( x + 7 ) ( x + 7 ) ( x + 2 )

Finally, add the numerators and simplify:

8 x + 41 ( x + 7 ) ( x + 2 )

Example 61

Express 5 x 7 − 3 x − 4 2 as a single fraction.

Solution

In this example both denominators are simply numbers. The lowest common denominator is 14, and both fractions are re-written with this denominator. Thus

5 x 7 − 3 x − 4 2 = 10 x 14 − 7 ( 3 x − 4 ) 14 = 10 x − 7 ( 3 x − 4 ) 14 = 28 − 11 x 14

Task!

Express 1 x + 1 y as a single fraction.

The simplest expression which has x and y as its factors is x y . This is the lowest common denominator. Both fractions are written using this denominator. Noting that 1 x = y x y and that 1 y = x x y we find

1 x + 1 y = y x y + x x y = y + x x y

No cancellation is now possible because neither x nor y is a factor of the numerator.

Exercises
  1. Simplify
    1. x 4 + x 7 ,
    2. 2 x 5 + x 9 ,
    3. 2 x 3 − 3 x 4 ,
    4. x x + 1 − 2 x + 2 ,
    5. x + 1 x + 3 x + 2 ,
    6. 2 x + 1 3 − x 2 ,
    7. x + 3 2 x + 1 − x 3 ,
    8. x 4 − x 5
  2. Find
    1. 1 x + 2 + 2 x + 3 ,
    2. 2 x + 3 + 5 x + 1 ,
    3. 2 2 x + 1 − 3 3 x + 2 ,
    4. x + 1 x + 3 + x + 4 x + 2 ,
    5. x − 1 x − 3 + x − 1 ( x − 3 ) 2 .
  3. Find 5 2 x + 3 + 4 ( 2 x + 3 ) 2 .
  4. Find 1 7 s + 11 21
  5. Express A 2 x + 3 + B x + 1 as a single fraction.
  6. Express A 2 x + 5 + B ( x − 1 ) + C ( x − 1 ) 2 as a single fraction.
  7. Express A x + 1 + B ( x + 1 ) 2 as a single fraction.
  8. Express A x + B x 2 + x + 10 + C x − 1 as a single fraction.
  9. Express A x + B + C x + 1 as a single fraction.
  10. Show that x 1 1 x 3 − 1 x 2 is equal to x 1 x 2 x 3 x 2 − x 3 .
  11. Find
    1. 3 x 4 − x 5 + x 3 ,
    2. 3 x 4 − x 5 + x 3 .
    1. 11 x 28 ,
    2. 23 x 45 ,
    3. − x 12 ,
    4. x 2 − 2 ( x + 1 ) ( x + 2 ) ,
    5. x 2 + 6 x + 2 x ( x + 2 ) ,
    6. x + 2 6 ,
    7. 9 + 2 x − 2 x 2 3 ( 2 x + 1 ) ,
    8. x 20
    1. 3 x + 7 ( x + 2 ) ( x + 3 ) ,
    2. 7 x + 17 ( x + 3 ) ( x + 1 ) ,
    3. 1 ( 2 x + 1 ) ( 3 x + 2 ) ,
    4. 2 x 2 + 10 x + 14 ( x + 3 ) ( x + 2 ) ,
    5. x 2 − 3 x + 2 ( x − 3 ) 2
  3. 10 x + 19 ( 2 x + 3 ) 2
  4. 3 s + 11 21
  5. A ( x + 1 ) + B ( 2 x + 3 ) ( 2 x + 3 ) ( x + 1 )
  6. A ( x − 1 ) 2 + B ( x − 1 ) ( 2 x + 5 ) + C ( 2 x + 5 ) ( 2 x + 5 ) ( x − 1 ) 2
  7. A ( x + 1 ) + B ( x + 1 ) 2
  8. ( A x + B ) ( x − 1 ) + C ( x 2 + x + 10 ) ( x − 1 ) ( x 2 + x + 10 )
  9. ( A x + B ) ( x + 1 ) + C x + 1
  10. Solution omitted
    1. 53 x 60 ,
    2. 13 x 60