Distributive Rule
A quantity outside a bracket multiplies each of the
terms inside the bracket. This is known as the distributive
rule.
Example 1
(a) 3( x  2 y ) = 3 x  6 y.
(b) 2 x ( x  2 y + z ) = 2 x^{ 2} 
4 xy + 2 xz.
(c) 7 y  4(2 x  3) = 7 y  8 x + 12 .
This is a relatively simple rule but, as in all mathematical
arguments, a great deal of care must be taken to proceed
correctly.
Exercise
Remove the brackets and simplify the following expressions.
(a) 5 x  7 x^{ 2}  (2 x )^{ 2}
(b) (3 y )^{ 2} + x^{ 2}  (2
y )^{ 2}
(c) 3 a + 2( a + 1)
(d) 5 x  2 x ( x  1)
(e) 3 xy  2 x ( y  2)
(f) 3 a ( a  4)  a ( a  2)
Solution
(a) First note that (2 x )^{ 2} = (2
x ) Ã— (2 x ) = 4 x^{ 2} .
5 x  7 x^{
2}  (2 x )^{ 2 } 
= 5 x  7 x^{ 2}  4 x^{ 2}
= 5 x  11 x^{ 2}

(b)
(3 y )^{ 2}
+ x^{ 2}  (2 y )^{ 2} 
= 9 y^{ 2} + x^{ 2}  4 y^{
2} = 9 y^{ 2}  4 y^{ 2} + x^{
2}
= 5 y^{ 2} + x^{ 2}

(c)
3 a + 2( a +
1) 
= 3 a + 2 a + 2 = 5 a + 2

(d)
5 x  2 x ( x
 1) 
= 5 x  2 x^{ 2} + 2 x = 7 x 
2 x^{ 2}

(e)
3 xy  2 x ( y
 2) 
= 3 xy  2 xy + 4 x = xy + 4 x

(f)
3 a ( a  4) 
a ( a  2) 
= 3 a^{ 2}  12 a  a^{ 2}
+ 2 a = 3 a^{ 2}  a^{ 2} + 2 a  12 a
= 2 a^{ 2 } 10 a

In the case of two brackets being multiplied
together, to simplify the expression first choose one
bracket as a single entity and multiply this into the other
bracket.
Example 2
For each of the following expressions, multiply out the
brackets and simplify as far as possible.
(a) ( x + 5)( x + 2) ,
(b) (3 x  2)(2 y + 3) .
Solution
( a )
( x + 5)( x +
2) 
= ( x + 5) x + ( x + 5)^{ 2} =
x( x + 5) + 2( x + 5)
= x^{ 2} + 5 x + 2 x + 10
= x^{ 2} + 7 x + 10 .

( b )
(3 x  2)(2 y
+ 3) 
= (3 x  2)2 y + (3 x  2) 3 = 2 y(3 x
 2) + 3(3 x  2)
= 6 xy  4 y + 9 x  6 .

Try this short quiz.
Quiz
To which of the following does the expression (2 x  1)( x +
4) simplify?
(a) 2 x^{ 2}  2 x + 4
(b) 2 x^{ 2}  7 x + 4
(c) 2 x^{ 2} + 7 x  4
(d) 2 x^{ 2} + 2 x  4
Solution:
(2x  1)(x +
4) 
= (2x  1)x + (2x  1)4 = (2x^{ 2}
 x) + (8x  4)
= 2x^{ 2}  x + 8x  4
= 2x^{ 2} + 7x  4

