1 Cancelling common factors

Consider the fraction 10 35 . To simplify it we can factorise the numerator and the denominator and then cancel any common factors. Common factors are those factors which occur in both the numerator and the denominator. Thus

10 35 = × 2 7 × = 2 7

Note that the common factor 5 has been cancelled. It is important to remember that only common factors can be cancelled. The fractions 10 35 and 2 7 have identical values - they are equivalent fractions - but 2 7 is in a simpler form than 10 35 .

We apply the same process when simplifying algebraic fractions.

Example 49

Simplify, if possible,

  1. y x 2 x ,
  2. x x y ,
  3. x x + y
Solution
  1. In the expression y x 2 x , x is a factor common to both numerator and denominator. This common factor can be cancelled to give

    y 2 = y 2

  2. Note that x x y can be written 1 x x y . The common factor of x can be cancelled to give

    1 y = 1 y

  3. In the expression x x + y notice that an x appears in both numerator and denominator. However x is not a common factor. Recall that factors of an expression are multiplied together whereas in the denominator x is added to y . This expression cannot be simplified.
Task!

Simplify, if possible,

  1. a b c 3 a c ,
  2. 3 a b b + a

    When simplifying remember only common factors can be cancelled.

  1. b 3
  2. This cannot be simplified.
Task!

Simplify 21 x 3 14 x ,

Factorising and cancelling common factors gives:

21 x 3 14 x = × 3 × × x 2 × 2 × = 3 x 2 2

Task!

Simplify 36 x 12 x 3

Factorising and cancelling common factors gives:

36 x 12 x 3 = 12 × 3 × x 12 × x × x 2 = 3 x 2

Example 50

Simplify 3 x + 6 6 x + 12 .

Solution

First we factorise the numerator and the denominator to see if there are any common factors.

3 x + 6 6 x + 12 = 3 ( x + 2 ) 6 ( x + 2 ) = 3 6 = 1 2

The factors x + 2 and 3 have been cancelled.

Task!

Simplify 12 2 x + 8 .

Factorise the numerator and denominator, and cancel any common factors. 6 × 2 2 ( x + 4 ) = 6 x + 4

Example 51

Show that the algebraic fraction 3 x + 1 and 3 ( x + 4 ) x 2 + 5 x + 4 are equivalent.

Solution

The denominator, x 2 + 5 x + 4 , can be factorised as ( x + 1 ) ( x + 4 ) so that

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

Note that ( x + 4 ) is a factor common to both the numerator and the denominator and can be cancelled to leave 3 x + 1 . Thus 3 x + 1 and 3 ( x + 4 ) x 2 + 5 x + 4 are equivalent fractions.

Task!

Show that x 1 x 2 3 x + 2 is equivalent to 1 x 2 .

First factorise the denominator:

( x 1 ) ( x 2 )

Now identify the factor common to both numerator and denominator and cancel this common factor:

1 x 2 . Hence the two given fractions are equivalent.

Example 52

Simplify 6 ( 4 8 x ) ( x 2 ) 1 2 x

Solution

The factor 4 8 x can be factorised to 4 ( 1 2 x ) . Thus

6 ( 4 8 x ) ( x 2 ) 1 2 x = ( 6 ) ( 4 ) ( 1 2 x ) ( x 2 ) ( 1 2 x ) = 24 ( x 2 )

Task!

Simplify x 2 + 2 x 15 2 x 2 5 x 3

First factorise the numerator and factorise the denominator:

( x + 5 ) ( x 3 ) ( 2 x + 1 ) ( x 3 )

Then cancel any common factors:

x + 5 2 x + 1

Exercises
  1. Simplify, if possible,
    1. 19 38 ,
    2. 14 28 ,
    3. 35 40 ,
    4. 7 11 ,
    5. 14 56
  2. Simplify, if possible,
    1. 14 21 ,
    2. 36 96 ,
    3. 13 52 ,
    4. 52 13
  3. Simplify
    1. 5 z z ,
    2. 25 z 5 z ,
    3. 5 25 z 2 ,
    4. 5 z 25 z 2
  4. Simplify
    1. 4 x 3 x ,
    2. 15 x x 2 ,
    3. 4 s s 3 ,
    4. 21 x 4 7 x 3
  5. Simplify, if possible,
    1. x + 1 2 ( x + 1 ) ,
    2. x + 1 2 x + 2 ,
    3. 2 ( x + 1 ) x + 1 ,
    4. 3 x + 3 x + 1 ,
    5. 5 x 15 5 ,
    6. 5 x 15 x 3 .
  6. Simplify, if possible,
    1. 5 x + 15 25 x + 5 ,
    2. 5 x + 15 25 x ,
    3. 5 x + 15 25 ,
    4. 5 x + 15 25 x + 1
  7. Simplify
    1. x 2 + 10 x + 9 x 2 + 8 x 9 ,
    2. x 2 9 x 2 + 4 x 21 ,
    3. 2 x 2 x 1 2 x 2 + 5 x + 2 ,
    4. 3 x 2 4 x + 1 x 2 x ,
    5. 5 z 2 20 z 2 z 8
  8. Simplify
    1. 6 3 x + 9 ,
    2. 2 x 4 x 2 + 2 x ,
    3. 3 x 2 15 x 3 + 10 x 2
  9. Simplify
    1. x 2 1 x 2 + 5 x + 4 ,
    2. x 2 + 5 x + 6 x 2 + x 6 .
    1. 1 2 ,
    2. 1 2 ,
    3. 7 8 ,
    4. 7 11 ,
    5. 1 4 .
    1. 2 3 ,
    2. 3 8 ,
    3. 1 4 ,
    4. 4
    1. 5,
    2. 5,
    3. 1 5 z 2 ,
    4. 1 5 z .
    1. 4 3 ,
    2. 15 x ,
    3. 4 s 2 ,
    4. 3 x
    1. 1 2 ,
    2. 1 2 ,
    3. 2,
    4. 3,
    5. x 3 ,
    6. 5
    1. x + 3 5 x + 1 ,
    2. x + 3 5 x ,
    3. x + 3 5 ,
    4. 5 ( x + 3 ) 25 x + 1
    1. x + 1 x 1 ,
    2. x + 3 x + 7 ,
    3. x 1 x + 2 ,
    4. 3 x 1 x ,
    5. 5 z 2
    1. 2 x + 3 ,
    2. 1 2 x + 1 ,
    3. 3 5 ( 3 x + 2 ) .
    1. x 1 x + 4 ,
    2. x + 2 x 2 .