2. 3. Factorising
What does Factorising mean?…
Very simply, factorising is the opposite of expanding.
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.
Let’s make sure we understand about Factors…
The key to successful factorising is understanding factors,
Just write down what each term means in full and spot the factors…
12a 12 × a
6 y2 6 × y × y
7 pq 2 7 × p × q × q
3. Example 1 Example 2
Factorise: 7a + 21 Factorise: 10 p + 15 pq
1. common factors in both numbers and letters? 1. common factors in both numbers and letters?
Numbers: 7 and 21 Highest Factor = 7 Numbers: 10 and 15 Highest Factor = 5
Letters: there are no letters in the 2nd term, so Letters: p and pq Highest Factor = p
we can’t take any letters outside the bracket!
2. So we have… 2. So we have…
7( ? + ? ) 5p ( ? + ? )
3.
3.
7 × ? = 7a a 5 p × ? = 10 p 2
7 × ? = 21 3 5 p × ? = 15 pq 3q
Which gives us… 7(a + 3) Which gives us… 5 p (2 + 3q)
4. Expand the answer (on paper or in your head) 4. Expand the answer (on paper or in your head)
to make sure you get the original question! to make sure you get the original question!
4. Example 3
Factorise: 24c 2 + 16c
1. common factors in both numbers and letters?
Numbers: 24 and 16 Highest Factor = 8
Letters: c2 and c Highest Factor = c
Remember: c2 is just c x c
2. So we have…
8c ( ? + ? )
3.
8c × ? = 24c 2 3c
8c × ? = 16c 2
Which gives us… 8c(3c + 2)
4. Expand the answer (on paper or in your head)
to make sure you get the original question!
5. Example 4 Example 5 – Nightmare!
Factorise: 18bc − 45b 2 Factorise: 18a 2b − 6ab + 30ab 2
1. common factors in both numbers and letters? 1. common factors in both numbers and letters?
Numbers: 18 and 45 Highest Factor = 9 Numbers: 18 6 and 30 Highest Factor = 6
Letters: b c and b2 Highest Factor = b Letters: a2 b , a b and a b2 Highest Factor = a
b
Remember: b2 is just b x b and b c is just b x c Remember: a2 b is just a x a x b and
a b2 is just a
2. So we have… xbxb
2. So we have…
9b ( ? − ? ) 6ab ( ? − ? − ? )
3. 3.
9b × ? = 18bc 2c 6ab × ? = 18a 2b 3a
9b × ? = 45b 2 5b 6ab × ? = 6ab 1
Which gives us… 9b(2c − 5b) 6ab × ? = 30ab 2 5b
Which gives us…
4. Expand the answer (on paper or in your head) 6ab(3a − 1 + 5b)
to make sure you get the original question!
4/5. Check for common factors and Expand the
answer to make sure you are correct!
6. Exercise Are you ready for
the answers ?
(a) Factorise 15x + 5
= 5(3x +1)
(b) Factorise 6y² - 3y
= 3y(2y – 1)
(c) Factorise a² + a
= a(a + 1)
(d) Factorise 18x² + 12x
6x(3x + 2)
=
(e) Factorise xy² - xy
xy(y – 1)
=
Click for the next question
7. A Difference Of Two Squares.
Consider what happens Now you try the
when you multiply out : example below:
( x + y ) ( x – y) Example.
Multiply out:
( 5 x + 7 y )( 5 x – 7 y )
=x - xy + xy - y
2 2
=x -y 2 2
Answer:
= 25 x 2 - 49 y 2
This is a difference of two
squares.
8. What Goes In The Box ?
Mutiply out:
(1) ( 3 x + 6 y ) ( 3 x – 6 y) (4) ( x – 11 y ) ( x + 11 y)
=9x2 – 36 y 2 = x2 – 121 y 2
(2) ( 2 x – 4 y ) ( 2 x + 4 y) (5) ( 7 x + 2 y ) ( 7 x – 2 y)
=4x2 – 16 y 2 = 49 x 2 – 4y2
(3) ( 8 x + 9 y ) ( 8 x – 9 y) (6) ( 5 x – 9 y ) ( 5 x + 9 y)
= 64 x 2 – 81 y 2 = 25 x 2 – 81 y 2
(3) ( 5 x – 7 y ) ( 5 x + 7 y) (7) ( 3 x + 9 y ) ( 3 x – 9 y)
=9x2 – 81 y 2
= 25 x 2
– 49 y 2
9. Factorising A Difference Of Two Squares.
By considering the brackets required to produce the following factorise the following
examples directly:
Examples
(1) x 2 - 9 (5) 4x 2 - 36
= 4 (x 2 - 9)
= ( x - 3 )( x + 3 )
= 4( x - 3 ) ( x + 3 )
(2) x 2 - 16 (6) 9x 2 - 16y 2
= ( x - 4 )( x + 4 ) = ( 3x - 4y ) ( 3x + 4y )
(3) x 2 - 25 (7) 100g 2 - 49k 2
= ( x - 5 ) ( x+ 5 ) = ( 10g – 7k ) ( 10g + 7k )
(4) x 2 - y 2 (8) 144d 2 - 36w 2
= 36 (4d 2 - w 2)
= ( x - y )( x + y )
= 36 ( 2d - w)( 2d + w )
10. Exercise
Factorise each of the following
(1) x 2 - 9
(2) x 2 - 16
(3) x 2 - 25
(4) x 2 - y 2
(5) 4x 2 - 36
(6) 9x 2 - 16y 2
(7) 100g 2 - 49k 2
(8) 144d 2 - 36w 2
