Mr Barton’s Maths Notes
Algebra
3. Factorising
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3. Factorising
What on earth does Factorising mean?…
Very simply, factorising is the opposite of what we did in the previous section – 2. Brackets
Factorising just means: putting back into brackets
How to Factorise
1. Look for the highest common factors in each term (they could be letters or numbers)
2. Place these common factors outside the bracket
3. Write down what is now left inside the bracket – ask yourself: what do I need to multiply
the term outside the bracket by to get my original term?
4. Check carefully that there are no more common factors in your bracket
5. Check your answer by expanding your brackets – it takes 2 seconds and it means you have
definitely got the question correct!
Let’s make sure we understand about Factors…
The key to successful factorising is understanding factors, and if it helps, why not just write
down what each term means in full and then it’s dead easy to spot the factors…
12a
12  a
6y 2
6  y  y
7 pq 2
7 p q q
Example 1
Example 2
7a  21
Factorise:
1. Okay, so we’re on the hunt for common
factors in both numbers and letters:
Numbers: 7 and
21
10 p  15 pq
Factorise:
Highest Factor = 7
Letters: there are no letters in the 2nd term, so
we can’t take any letters outside the bracket!
1. Okay, so we’re on the hunt for common
factors in both numbers and letters:
Numbers: 10 and
Letters: p and
15
pq
Highest Factor = 5
Highest Factor = p
2. So we have…
2. So we have…
3. Now we have to figure out…
3. Now we have to figure out…
7( ?  ? )
7  ?  7a
7  ?  21
Which gives us…
a
3
7(a  3)
5p ( ?  ? )
5 p  ?  10 p
5 p  ?  15 pq
Which gives us…
2
3q
5 p (2  3q)
4. Check there are no more common factors left
inside the bracket…erm… nothing is common to
both a and 3, so we’re fine!
4. Check there are no more common factors left
inside the bracket…erm… nothing is common to
both 2 and 3q, so we’re fine!
5. Expand the answer (on paper or in your head)
to make sure you get the original question!
5. Expand the answer (on paper or in your head)
to make sure you get the original question!
Example 3
NOTE:
24c 2  16c
Factorise:
A very common mistake is not to take out the
highest common factor.
1. Okay, so we’re on the hunt for common
factors in both numbers and letters:
Numbers: 24 and
Letters: c2 and
16
Highest Factor = 8
c
Highest Factor = c
Remember:
c2
is just c x c
Numbers: 24 and
Letters: c2 and
16
c
Highest Factor = 2
Highest Factor = c
We would get…
2. So we have…
2c ( ?  ? )
8c ( ?  ? )
And then…
3. Now we have to figure out…
8c  ?  24c
8c  ?  16c
2
Which gives us…
For example, imagine we were doing Example 3,
but for the numbers we thought the highest
common factor was 2…
3c
2
8c(3c  2)
4. Check there are no more common factors
5. Expand the answer (on paper or in your head)
to make sure you get the original question!
2c  ?  24c2
2c  ?  16c
Which gives us…
12c
8
2c(12c  8)
But, so long as we remember to always check
there are no more common factors, we’ll be fine,
because a quick glance at this answers shows us
that 12 and 8 have a common factor of 4!
Example 5 – Nightmare!
Example 4
Factorise:
18bc  45b2
Factorise:
1. Okay, so we’re on the hunt for common
factors in both numbers and letters:
Numbers: 18 and
Letters: b c and
45
b2
Numbers: 18 6 and 30
Highest Factor = 6
Highest Factor = b
Letters: a2 b a b and a b2
Highest Factor = a b
2. So we have…
Remember: a2 b is just a x a x b and
a b2 is just a x b x b
2. So we have…
6ab ( ?  ?  ? )
9b ( ?  ? )
3. Now we have to figure out…
3. Now we have to figure out…
Which gives us…
1. Okay, so we’re on the hunt for common
factors in both numbers and letters:
Highest Factor = 9
Remember: b2 is just b x b and b c is just b x c
9b  ?  18bc
9b  ?  45b2
18a2b  6ab  30ab2
2c
5b
9b(2c  5b)
6ab  ?  18a 2b
6ab  ?  6ab
3a
1
6ab  ?  30ab2
5b
4. Check there are no more common factors
Which gives us…
5. Expand the answer (on paper or in your head)
to make sure you get the original question!
4/5. Check for common factors and Expand the
answer to make sure you are correct!
6ab(3a  1  5b)
Good luck with
your revision!