WJEC MATHEMATICS
INTERMEDIATE
ALGEBRA
EXPANDING
1
Contents
Single Brackets (Santa's hat)
Double Brackets (Foil)
Multiple brackets with simplifying
Credits
WJEC Question bank
http://www.wjec.co.uk/question-bank/question-search.html
2
Expanding brackets means that you want to remove brackets from
an expression. There are two main types;
Remember:
𝑥 + 𝑥 = 2𝑥
and
𝑥 × 𝑥 = 𝑥2
Single Brackets
A single bracket question looks like this:
3(2𝑥 + 4)
Which means 3 multiplied by 2𝑥 and multiplied by 4. So, to expand
this, we need to multiply the 3 and the 2𝑥 and then multiply the 3
and the 4. We can add multiply lines to show this:
3(2𝑥 + 4)
Look familiar?
This method is sometimes called the Santa's hat method
So, the answer to the above example:
6𝑥 + 12
3 × 2𝑥
3×4
3
Exercise A7
1.
5(6𝑦 + 2)
2.
7(3𝑎 + 8)
3.
4(3𝑥 + 3)
4.
12(9𝑐 + 8)
5.
𝑥(6𝑥 + 8)
Negative Numbers
Most marks are lost on these questions when people become
confused with negative signs. Consider this example:
2(5𝑥 − 7)
= 10𝑥 − 14
2 × 5𝑥
2 × −7
Consider this more difficult example:
−4(9𝑥 − 7)
Remember: Multiplying two
negative numbers results in
= −36𝑥 + 28
−4 × 9𝑥
a positive number
−4 × −7
4
Exercise A8
Expand:
1.
4(3𝑦 − 2)
2.
9(5𝑎 − 2)
3.
−5(8𝑥 − 11)
4.
−12(8𝑐 − 11)
5.
−𝑥(4𝑥 − 0.5)
Double Brackets
Double brackets questions typically look like this:
(𝑥 + 3)(𝑥 + 5)
To expand this, we need to multiply every term in the first bracket by
every term in the second bracket. We can draw multiplication lines
on as follows:
(𝑥 + 3)(𝑥 + 5)
We need to multiply the first term in each bracket, the outside term
of each bracket, the inside term in each bracket and the last term in
each bracket.
FIRST - OUTSIDE - INSIDE - LAST
(Or FOIL for short)
5
(𝑥 + 3)(𝑥 + 5)
FIRST
(𝑥 + 3)(𝑥 + 5)
OUTSIDE
(𝑥 + 3)(𝑥 + 5)
INSIDE
(𝑥 + 3)(𝑥 + 5)
LAST
Going back to the earlier example:
(𝑥 + 3)(𝑥 + 5)
Expanding, we get:
𝑥 2 + 5𝑥 + 3𝑥 + 15
FIRST
𝑥×𝑥
OUTSIDE
𝑥×5
INSIDE
3×𝑥
LAST
3×5
Notice how the middle two terms can be simplified. If you are unsure
of this, see the Simplifying booklet.
So, we get
𝑥 2 + 8𝑥 + 15
Again, take care with negative signs
(𝑥 − 4)(𝑥 − 7)
= 𝑥 2 − 7𝑥 − 4𝑥 + 28
FIRST
𝑥×𝑥
OUTSIDE
𝑥 × −7
INSIDE
−4 × 𝑥
LAST
−4 × −7
6
= 𝑥 2 − 11𝑥 + 28
Exercise A9
Expand:
1.
(𝑥 + 4)(𝑥 + 9)
2.
(𝑥 + 3)(𝑥 + 8)
3.
(𝑥 + 1)(𝑥 + 4)
4.
(𝑥 + 3)(𝑥 + 7)
5.
(𝑥 + 2)(𝑥 + 8)
6.
(𝑥 + 9)(𝑥 + 4)
7.
(𝑥 + 2)(𝑥 + 5)
8.
(𝑥 + 6)(𝑥 + 2)
9.
(𝑥 + 3)(𝑥 + 8)
10.
(𝑥 − 4)2
Remember: 'Squared'
means multiplied by
itself
7
Multiple Brackets and Simplifying
Lots of questions at GCSE require you to expand brackets and then
simplify. If you get stuck on the simplifying step, see the booklet
'Simplifying'.
Example
Consider this question. It's clearly a question that required two
Santa's hats!
2(4𝑥 − 3) − 7(2𝑥 + 5)
= 8𝑥 − 6 − 14𝑥 − 35
2×4𝑥
2×−3
−7×2𝑥
−7×+5
= −6𝑥 − 41
Example 2 (Secret Santa's hat)
On first glance, this question does not seem like a Santa's hat
question but there is a sign in front of the second bracket. This minus
sign acts on the 2𝑥 and on the −8𝑦. We can show this with
multiplication lines.
(4𝑥 + 2𝑦) − (2𝑥 − 8𝑦)
= 4𝑥 + 2𝑦 − 2𝑥 + 8𝑦
This simplifies to;
−(−8)
= 2𝑥 + 10𝑦
8
Exercise A10
Expand and simplify:
1.
2(9𝑥 − 3) − 5(6𝑥 + 9)
2.
7(12𝑦 − 4) − 4(2𝑦 + 11)
3.
4(8𝑎 − 6) − 7(9𝑎 + 7)
4.
11(6𝑥 − 9𝑦) − (𝑥 + 2𝑦)
5.
9(6𝑥 − 𝑏) − (7𝑥 + 9𝑏)
Exam Questions A3
1.
2.
3.
4.
9
5.
6.
7.
8.
9.
10