Factorising
Reminder – multiplying out brackets.
Remember to multiply all parts inside the brackets by the number or letter
outside.
1. 5(2p – 4) = 5×2p – 5×4
= 10p – 20
2. a( 3b + c) = a×3b + a×c
= 3ab + ac
Multiply and simplify
1. 6(2d + 4) – 17
= 12d + 24 – 17
= 12d + 7
2. -3 + 4(7 – f)
= -3 + 28 – 4f
= 25 -4f
Factors
The factors of a number are all the numbers that divide into it exactly.
(Remember Factor Family go into Multiple Mansion). The number 1 and the
number itself are always factors.
Factors of
15 are
20 are
36 are
1, 3, 5, 15
1, 2, 4, 5, 10, 20
1, 2, 3, 4, 6, 9, 12, 36
Highest common factor – look for the highest common factor that appears in
both numbers.
12 and 20 - Factors of 12 are 1, 2, 3, 4, 6, 12
Factors of 20 are 1, 2, 4, 5, 10, 20
Highest common factor is 4
Factorising – is the opposite of multiplying out brackets.
Look for a common factor and then that number goes outside the bracket.
Ex1
4d + 32
= 4(d + 8)
4 is the highest common factor
Ex2
12e – 10
= 2(6e – 5)
2 is the highest common factor
Ex 3
24p + 36q
= 12(2p +3q)
12 is the highest common factor
Ex 4
5a – 15b + 30
= 5(a – 3b + 6)
5 is the highest common factor
Ex 5
pq + pr
= p(q +r)
p is the common factor
Ex 6
6x2 – 9xy
= 3x (2x – 3y)
3x is the common factor
Ex 7
Expand, simplify then factorise
2(m + 3) + 5(m – 11)
= 2m + 6 + 5m – 55
= 7m – 49
7 is the common factor
= 7(m – 7)
Difference of two squares
Ex 1
x2 – y2 = (x – y)(x + y)
Ex 2
a2 – 16 = (a – 4)(a + 4)
Ex 3
64 – y2 = (8 – y)(8 + y)
Ex 4
4x2 – 25y2 = (2x – 5y)(2x + 5y)
Ex 5
532 – 472 = (53 -47)(53 + 47) = 6 x 100 = 600
Ex 6
9y2 – 16x2 =
Ex 7
13.52 – 3.52 =
Common factors and a Difference of Two squares
Take out the common factor then factorise
Ex 1
3x2 – 12
= 3(x2 – 4)
= 3(x – 2)(x + 2)
Ex 2
36 – 4y2
= 4(9 – y2)
= 4(3 – y)(3 + y)
Ex 3
20y2 – 5
= 5(4y2 – 1)
= 5(2y – 1)(2y + 1)
Ex 4
2de2 – 2df2
= 2d(e2 – f2)
= 2d(e – f)(e + f)
Ex 5
2x2 – 32y2
=
Ex 6
12kx2 – 3ky2
=
Factorising a Quadratic
A quadratic expression contains an x2 term.
A trinomial is an expression containing 3 terms
To factorise find two numbers that multiply to make the last number and add to
make the middle number.
Ex 1
x2 + 9x + 20
= (x + 5)(x + 4)
5 x 4 = 20
5+4=9
Ex 2
x2 – 11x + 24
= (x – 3)(x – 8)
-3 x -8 = 24
-3 + -8 = -11
Ex 3
x2 + 3x – 4
= (x - 1)(x + 4)
-1 x 4 = -4
-1 + 4 = -3
Ex 4
x2 – x – 30
= (x + 5)(x – 6)
5 x -6 = -30
5 + (-6) = -1
Ex 5
y2 + 8y + 15
=
Ex 6
u2 – 11u + 18
=
Ex 7
b2 + 8b – 20
=
Ex 8
n2 – n – 12
=
Factorising a Quadratic where coefficient of x2 is greater than 1
Use borrow and payback method
Ex 1
3x2 + 7x + 4
Borrow(3 x 4)
Factorise
Payback(3 to the x in each bracket)
Take out any common factors
Cross out common factor
12x2 – 23x +10
Ex 2
Borrow(12 x 10)
Factorise
Payback(12 to the x in each bracket)
Take out any common factors
Cross out common factor
Ex 3
B (6 x -5)
F
P (6)
CF
6x2 + 7x – 5
4x2 – 5x – 9
Ex 4
B (4 x -9)
F
P (4)
CF
Ex 5
4x2 – 8x + 3
=
= x2 + 7x + 12
= (x + 3)(x + 4)
= (3x + 3)(3x +4)
= 3(x + 1)(3x + 4)
= (x + 1)(3x + 4)
= x2 – 23x +120
= (x – 8)(x – 15)
= (12x – 8)(12x -15)
= 4(3x – 2)3(4x – 5)
= (3x – 2)(4x – 5)
= x2 + 7x -30
= (x + 10)(x - 3)
= (6x + 10)(6x – 3)
= 2(3x + 5)3(2x – 1)
= (3x + 5)(2x – 1)
= x2 - 5x – 36
= (x + 4)(x - 9)
= (4x + 4)(4x – 9)
= 4(x + 1)(4x -9)
= (x + 1)(4x – 9)
Ex 6
6a2 + 17a - 3
=
We now have 3 ways of factorising –
1. Common factor
2. Difference of two squares
3. Quadratic
Always check for a common factor before factorising
Ex 1
5x2 + 35
= 5(x2 + 7)
Ex 2
3x2 – 12
= 3(x2 – 4)
= 3(x – 2)(x + 2)
Ex 3
6x2 - 18x - 24
= 6(x2 – 3x -4)
= 6 (x - 4)(x + 1)
Ex 4
9x2 + 24x + 12
= 3(3x2 + 8x + 4)
= 3(x2 + 8x + 12)
= 3(x + 6)(x + 2)
= 3(3x + 6)(3x + 2)
= 3 [3(x + 2)(3x + 2)]
= 3(x + 2)(3x + 2)
Ex 5
t2 – tu
=
Ex 6
72 – 2m2
=
Ex 7
2n2 – 2n – 144
=
Ex 8
12n2 - 33n + 18
=
B
F
P
CF
You can always check your factorising answer by multiplying out the brackets.