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Syllabus Edition
First teaching 2021
Last exams 2024
Factorising (CIE IGCSE Maths: Extended)
Revision Note
Basic Factorising
What is factorisation?
- A factorised expression is one written as the product (multiplication) of two, or more, terms (factors)
- 3(x + 2) is factorised, as it is 3 × (x + 2)
- 3x + 6 is not factorised as it is "something" + "something"
- 3xy is factorised as it is 3 × x × y
- 12 can also be factorised: 12 = 2 x 2 x 3
- In algebra, factorisation is the opposite of expanding brackets
- it's "putting it into" brackets
How do I factorise two terms?
- To factorise 12x2 + 18x
- The highest common factor of 12 and 18 is 6
- The highest common factor of x2 and x is x
- this is the largest letter that divides both x2 and x
- Multiply both to get the common factor
- 6x
- Rewrite each term in 12x2 + 18x as "common factor × something"
- 6x × 2x + 6x × 3
- "Take out" the common factor by writing it outside brackets
- Put the remaining 2x + 3 inside the brackets
- Answer: 6x(2x + 3)
- Check this expands to give 12x2 + 18x
Examiner Tip
- You can always check that your factorisation is correct by simply expanding the brackets in your answer!
Worked example
Factorise 5x + 15
Find the highest common factor of 5 and 15
5
There is no x in the second term, so no highest common factor in x needed
Write each term as 5 × "something"
5 × x + 5 × 3
"Take out" the 5
5(x + 3)
5(x + 3)
Write each term as 6x × "something"
6x × 5x - 6x × 4
"Take out" the 6x
6x(5x - 4)
6x(5x - 4)
Factorising by Grouping
How do I factorise expressions with common brackets?
- To factorise 3x(t + 4) + 2(t + 4), both terms have a common bracket, (t + 4)
- the whole bracket, (t + 4), can be "taken out" like a common factor
- (t + 4)(3x + 2)
- this is like factorising 3xy + 2y to y(3x + 2)
- y represents (t + 4) above
- the whole bracket, (t + 4), can be "taken out" like a common factor
How do I factorise by grouping?
- Some questions may require you to form the common bracket yourself
- for example, factorise xy + px + qy + pq
- "group" the first pair of terms, xy + px, and factorise, x(y + p)
- "group" the second pair of terms, qy + pq, and factorise, q(y + p),
- now factorise x(y + p) + q(y + p) as above
- (y + p)(x + q)
- for example, factorise xy + px + qy + pq
-
- This is called factorising by grouping
- The groupings are not always the first pair of terms and the second pair of terms, but two terms with common factors
Examiner Tip
- As always, once you have factorised something, expand it by hand to check your answer is correct.
Worked example
Factorise ab + 3b + 2a + 6
Method 1
Notice that ab and 3b have a common factor of b
Notice that 2a and 6 have a common factor of 2
Factorise the first two terms, using b as a common factor
b(a + 3) + 2a + 6
Factorise the second two terms, using 2 as a common factor
b(a + 3) + 2(a + 3)
(a + 3) is a common bracket
We can factorise using (a + 3) as a factor
(a + 3)(b + 2)
Method 2
Notice that ab and 2a have a common factor of a
Notice that 3b and 6 have a common factor of 3
Rewrite the expression grouping these terms together
ab + 2a + 3b + 6
Factorise the first two terms, using a as a common factor
a(b + 2) + 3b + 6
Factorise the second two terms, using 3 as a common factor
a(b + 2) + 3(b + 2)
(b + 2) is a common bracket
We can factorise using (b + 2) as a factor
(b + 2)(a + 3)
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