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 ) × (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
|
|
