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Grouping symbols

There are two laws for grouping symbols – the distributive law and the factorisation law. We look at both these laws for grouping symbols in greater detail below:

 

Distributive law for grouping symbols

This law is used to expand an expression with a grouping symbol. We multiply the term outside the grouping symbol by each term inside the grouping.

For example: a(b + c) = a x b + a x c

= ab + ac.

Here ‘a’ is the term outside the group, and ‘b + c’ is the grouped term. When we expand the expression, we multiply ‘a’ by both ‘b’ and ‘c’ (they are inside the group).

Similarly 3(x – 2) = 3 x x – 3 x 2

= 3x – 6

In general a(b – c) = ab – ac, and a(b + c) = ab + ac.

Now, let’s look at some more examples of applying the distributive law, and simplifying the expressions below:

  1. 3(x – 3) = 3x – 9
  2. 3(2x – 3) = 6x – 9
  3. 4(2x + 3) + 2(-x + 3)=  8x + 12 + (-2x) + 6

    =  8x – 2x + 12 + 6

    = 6x + 18

  4. 3x(x + 2) – 2(x2 + 2)=  3x2 + 6x – 2x2 – 4

    =  3x2 – 2x2 + 6x – 4

    =  x2 + 6x – 4

 

Factorisation law for grouping symbols

This is the opposite of Distributive law. When we apply factorisation, we take common terms and common numerical value (highest common factor) out of the expression, and group the algebraic expression.

Let’s look at some examples:

  1. 9x + 3 = 3(3x + 1)
  2. 9x + 3y = 3(3x + y)
  3. -3xy + 6x  =  3x(-y + 2)
  4. 6x2y + 3xz  =  3x(2xy + z)
  5. -36xy – 9yz  =  -9y(4x + z)