2. Before attempting this sheet, students should be able to:
• Use algebraic vocabulary.
• Calculate factors and highest common factors.
• Expand brackets.
Prior Knowledge:
3. Factorising is the inverse (opposite) of multiplying out / expanding brackets. You
have to look for common factors – numbers or letters (or both) that go into every
term.
Example 1
Fully factorise 10x + 15.
Step 1.
Find the highest common factor for the expression. In this case, it is 5, because 5 is
the highest number which will divide into both 10x and 15. Be careful - if you choose
a common factor which is not the highest common factor, your final answer will not
be fully factorised.
Step 2.
Write the highest common factor outside of the bracket:
5( )
4. Example 1
Fully factorise 10x + 15.
Step 3.
Now, find the terms which go inside the bracket. To do this, divide each term in the
original expression by the term outside the brackets (remember, we multiply to
expand brackets and divide to factorise).
10x ÷ 5 = 2x
15 ÷ 5 = 3
The final answer is 5(2x + 3).
Step 4.
Always check your answer by multiplying out the bracket to see if it matches the
original expression:
5 × 2x = 10x
5 × 3 = 15
This gives us 10x + 15, therefore 5(2x + 3) is correct.
5. Sometimes the highest common factor may include a number and a letter.
Example 2
Factorise 6x2 – 9xy.
Step 1.
3 is the highest common factor of 6 and 9, and x is common to both x2 and xy.
Combine these to get the term outside the brackets: 3x.
Step 2.
Place the common factor on the outside of the bracket.
3x( )
6. Example 2
Factorise 6x2 – 9xy.
Step 3.
Divide each term in the original expression by 3x to find the terms inside the bracket
(watch out for the second term – as it is negative):
6x2 ÷ 3x = 2x
-9xy ÷ 3x = -3y
Put these into the brackets to get the final answer:
3x(2x – 3y)
Step 4.
Expand the bracket to check your answer.
3x × 2x = 6x2
3x × -3y = -9xy
This gives us 6x2 – 9xy, therefore the answer is correct.
8. 1. Write the highest common factor of the two terms in each expression:
Your turn:
2. Fill in the gaps:
a. 2x + 6
b. 8x + 6
c. 5x – 10
d. 4x + 16
e. 3x2 + 9x
a. 7x + 28 = (x + 4)
b. 12x + 18 = 6( + 3)
c. 15x – 10 = 5(3x )
d. 24x + 18 = (4x + 3)
e. 14x – 21 = 7( )
f. 6x + 4 = ( )
3. Fully factorise each expression:
a. 10x + 5
b. 2x – 8
c. 5m – 25
d. 6x + 3
e. 10t + 30
f. 7a – 14
g. 3y + 12
h. 8z – 10
9. 4. Fully factorise each of the following expressions.
Hint: look for two common factors.
Your turn:
5. Each of the expressions below have been fully factorised. Some are correct and
some are not. For each question, say whether it is correct. If it is wrong, explain
what mistake has been made and correct it.
a. 2x2 + 6x
b. 2y2 – 8y
c. 9x2 + 3xy
d. 4ab – 6bc
e. 4q2 – 8pq
f. 2y3 + 4y
a. 5x2 + 10x = 5x(x + 2)
b. 24x + 36 = 6(4x + 6)
c. 7x2 – 14xy = 7x(x + 2y)
d. 2xy – 4y = 2y(x – 4y)
e. 18x + 3 = 3(1 + 6x)
Challenge
Explain why 8x + 3y cannot be factorised.
10. 1. Write the highest common factor of the two terms in each expression:
2. Fill in the gaps:
a. 2x + 6
2
b. 8x + 6
2
c. 5x – 10
5
d. 4x + 16
4
e. 3x2 + 9x
3x
a. 7x + 28 = 7(x + 4)
b. 12x + 18 = 6(2x + 3)
c. 15x – 10 = 5(3x – 2)
d. 24x + 18 = 6(4x + 3)
e. 14x – 21 = 7(2x – 3)
f. 6x + 4 = 2(3x + 2)
Answers:
11. 3. Fully factorise each expression:
a. 10x + 5
5(2x + 1)
b. 2x – 8
2(x – 4)
c. 5m – 25
5(m – 5)
d. 6x + 3
3(2x + 1)
e. 10t + 30
10(t + 3)
f. 7a – 14
7(a – 2)
g. 3y + 12
3(y + 4)
h. 8z – 10
2(4z – 5)
Answers:
12. 4. Fully factorise each of the following expressions.
Hint: look for two common factors.
a. 2x2 + 6x
2x(x + 3)
b. 2y2 – 8y
2y(y – 4)
c. 9x2 + 3xy
3x(3x + y)
d. 4ab – 6bc
2b(2a – 3c)
e. 4q2 – 8pq
4q(q – 2p)
f. 2y3 + 4y
2y(y2 + 2)
Answers:
13. 5. Each of the expressions below have been fully factorised. Some are correct and
some are not. For each question, say whether it is correct. If it is wrong, explain
what mistake has been made and correct it.
a. 5x2 + 10x = 5x(x + 2)
Correct
b. 24x + 36 = 6(4x + 6)
Wrong – this is factorised, but not fully factorised – the terms inside the
bracket have a common factor of 2.
12(2x + 3)
c. 7x2 – 14xy = 7x(x + 2y)
Wrong – the second term in the bracket should be negative.
7x(x – 2y)
Answers:
14. d. 2xy – 4y = 2y(x – 4y)
Wrong – the second term in the bracket has not been divided by 2y.
2y(x – 2)
e. 18x + 3 = 3(1 + 6x)
This is correct, although the order of the terms has been reversed.
Challenge
Explain why 8x + 3y cannot be factorised.
There are no common factors between the terms other than 1.
Answers:
16. This game is designed for two players – each player needs a grid.
Each player starts by drawing the following ships on their grids (make sure they’re
hidden from the other player):
One 4-square long ship.
Two 3-square long ships.
Two 2-square long ships.
Ships may be placed horizontally or vertically, but not diagonally.
The first player then chooses a location on the grid to shoot.
When targeting a location, a player must create and fully factorise an expression,
formed from the row and column headings of the square they wish to shoot.
Mastery Task:
17. For example, to target the bottom-right square: the column heading is 2xy and the
row heading is -40. This factorises to 2(xy – 20); the player would use this expression
to target the location.
The other player then expands the brackets to 2xy and -40, checks that location and
says whether the shot hit or missed. If the targeted location has been incorrectly
factorised, the shot automatically misses, whether a ship is present or not.
The second player then takes a turn to shoot.
A ship is sunk when each square it is in has been hit.
A player wins when all their opponents ships have been sunk.
Hint: As you make your shots, label them on your own grid as hits or misses.
Mastery Task:
