Expanding Brackets Expanding brackets is a quicker way of calculating what an expression which is written several times would simplify down to.
e.g 3y + 7 + 3y + 7 + 3y + 7 + 3y + 7 + 3y + 7 + 3y + 7 + 3y + 7 + 3y + 7
this can be simplified, however a quicker way would be to expand the expression (multiply it out!)
Expand x
8(3y + 7) When expanding brackets we MULTIPLY everthing outside the bracket by everything inside the bracket in turn.
x
= 24y + 56
so: 8 times 3y = 24y
8 times 7 = 56
We can ONLY multiply numbers with other numbers
1
The next step is to multiply a negative term through the bracket
e.g.2 Expand
­4(3m ­ 5)
= ­12m + 20
We use the same method as before but need to remember:
pos x pos = pos
neg x neg = pos
neg x pos = neg
pos x neg = neg
2
Once you have grasped how to multiply a negative into a bracket the next step is to multiply an unknown into a bracket
e.g expand: y( y + 3 )
we use the same method as before and multiply everything outside the bracket by everything inside the bracket in turn
x
y( y + 3 )
x
y x y = y2
y x 3 = 3y
= y2 + 3y
3
Another more complicated example of multiplying an unknown term into a bracket would be:
expand: 3m( 4m ­ 7 )
We use the same method as before of multiplying the term outside the bracket by each term inside the bracket in turn.
This gives us
x
3m( 4m ­ 7 )
x
When multiplying terms in algebra you must multiply numbers and letters seperately!!
This gives us: 3m x 4m = 12m2 (3x4=12, m x m = m2)
3m x ­7 = ­21m (3 x ­7 = ­21, the m stays as
it is as there is no other m
to multiply it by)
= 12m2 ­ 21m
4
A commomn question is to expand and then simplify brackets
e.g expand and then simplify 5(m + 7) + 3(4m ­ 6)
To do this you need to expand each bracket seperately and join the answers together to make one longer answer
x
x
5(m + 7) + 3(4m ­ 6)
x
x
= 5m + 35 + 12m ­18 now we simplify this (see simplifying expressions for info on this)
5m + 35 + 12m ­18
= 5m + 12m + 35 ­ 18
= 17m + 17
5
Sometimes you can get asked to multiply two bracketed expressions together
What you need to remember is to multiply every term in one bracket by every term in the other bracket in turn
e.g calculate: ( 2y + 3 )( 3y ­ 2 )
We have 2 terms in each bracket so we have 4 multiplications to do
x
x
( 2y + 3 )( 3y ­ 2 )
This gives us:
x
x
2y x 3y = 6y2
3 x 3y = 9y
2y x ­ 2 = ­4y
3 x ­2 = ­6
We put these 4 multiplications together to form an expression
6y2 + 9y ­ 4y ­ 6 We can simplify this to get 6y2 + 5y ­6 6
Factorising is the inverse (opposite) to expanding a bracket.
It involves finding a common factor that runs through each term in your expression and dividing each term by it.
e.g Factorise: 8x2 + 12
There are 2 tems in this expression and if we look at the number part to them both we can see that both 8 and 12 can be divided by 4.
Dividing all terms by 4 gives us: 2x2 and 3
However we need to take the 4 that we divided into both terms and put it outside some brackets like so:
4( 2x2 + 3)
7