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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

 

 
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