1 Addition and subtraction of like terms

Like terms are multiples of the same quantity. For example 5 y , 17 y and 1 2 y are all multiples of y and so are like terms. Similarly, 3 x 2 , − 5 x 2 and 1 4 x 2 are all multiples of x 2 and so are like terms.

Further examples of like terms are:

k x and ℓ x which are both multiples of x ,

x 2 y , 6 x 2 y , − 13 x 2 y , − 2 y x 2 , which are all multiples of x 2 y

a b c 2 , − 7 a b c 2 , k a b c 2 , are all multiples of a b c 2

Like terms can be added or subtracted in order to simplify expressions.

Example 27

Simplify 5 x − 13 x + 22 x .

Solution

All three terms are multiples of x and so are like terms. The expression can be simplified to 14 x .

Example 28

Simplify 5 z + 2 x .

Solution

5 z and 2 x are not like terms. They are not multiples of the same quantity. This expression cannot be simplified.

Task!

Simplify 5 a + 2 b − 7 a − 9 b .

− 2 a − 7 b

Example 29

Simplify 2 x 2 − 7 x + 11 x 2 + x .

Solution

2 x 2 and 11 x 2 , both being multiples of x 2 , can be collected together and added to give 13 x 2 .

Similarly, − 7 x and x can be added to give − 6 x .

We get 2 x 2 − 7 x + 11 x 2 + x = 13 x 2 − 6 x which cannot be simplified further.

Task!

Simplify 1 2 x + 3 4 x − 2 y .

5 4 x − 2 y

Example 30

Simplify 3 a 2 b − 7 a 2 b − 2 b 2 + a 2 .

Solution

Note that 3 a 2 b and 7 a 2 b are both multiples of a 2 b and so are like terms. There are no other like terms. Therefore

3 a 2 b − 7 a 2 b − 2 b 2 + a 2 = − 4 a 2 b − 2 b 2 + a 2

Exercises
  1. Simplify, if possible,
    1. 5 x + 2 x + 3 x ,
    2. 3 q − 2 q + 11 q ,
    3. 7 x 2 + 11 x 2 ,
    4. − 11 v 2 + 2 v 2 ,
    5. 5 p + 3 q
  2. Simplify, if possible,
    1. 5 w + 3 r − 2 w + r ,
    2. 5 w 2 + w + 1 ,
    3. 6 w 2 + w 2 − 3 w 2
  3. Simplify, if possible,
    1. 7 x + 2 + 3 x + 8 x − 11 ,
    2. 2 x 2 − 3 x + 6 x − 2 ,
    3. − 5 x 2 − 3 x 2 + 11 x + 11 ,
    4. 4 q 2 − 4 r 2 + 11 r + 6 q ,
    5. a 2 + b a + a b + b 2 ,
    6. 3 x 2 + 4 x + 6 x + 8 ,
    7. s 3 + 3 s 2 + 2 s 2 + 6 s + 4 s + 12 .
  4. Explain the distinction, if any, between each of the following expressions, and simplify if possible.
    1. 18 x − 9 x ,
    2. 18 x ( 9 x ) ,
    3. 18 x ( − 9 x ) ,
    4. − 18 x − 9 x ,
    5. − 18 x ( 9 x )
  5. Explain the distinction, if any, between each of the following expressions, and simplify if possible.
    1. 4 x − 2 x ,
    2. 4 x ( − 2 x ) ,
    3. 4 x ( 2 x ) ,
    4. − 4 x ( 2 x ) ,
    5. − 4 x − 2 x ,
    6. ( 4 x ) ( 2 x )
  6. Simplify, if possible,
    1. 2 3 x 2 + x 2 3 ,
    2. 0.5 x 2 + 3 4 x 2 − 11 2 x ,
    3. 3 x 3 − 11 x + 3 y x + 11 ,
    4. − 4 α x 2 + β x 2 where α and β are constants.
    1. 10 x ,
    2. 12 q ,
    3. 18 x 2 ,
    4. − 9 v 2 ,
    5. cannot be simplified.
    1. 3 w + 4 r ,
    2. cannot be simplified,
    3. 4 w 2
    1. 18 x − 9 ,
    2. 2 x 2 + 3 x − 2 ,
    3. − 8 x 2 + 11 x + 11 ,
    4. cannot be simplified,
    5. a 2 + 2 a b + b 2 ,
    6. 3 x 2 + 10 x + 8 ,
    7. s 3 + 5 s 2 + 10 s + 12
    1. 9 x ,
    2. 162 x 2 ,
    3. − 162 x 2 ,
    4. − 27 x ,
    5. − 162 x 2
    1. 4 x − 2 x = 2 x ,
    2. 4 x ( − 2 x ) = − 8 x 2 ,
    3. 4 x ( 2 x ) = 8 x 2 ,
    4. − 4 x ( 2 x ) = − 8 x 2 ,
    5. − 4 x − 2 x = − 6 x ,
    6. ( 4 x ) ( 2 x ) = 8 x 2
    1. x 2 ,
    2. 1.25 x 2 − 11 2 x ,
    3. cannot be simplified,
    4. ( β − 4 α ) x 2