1. FACTORISING 2

2. REMINDER:

3. Factorising expressions of the form x2 + bx + c
Factorising is the reverse of expanding.
EXPAND
3 + 7
3 × 7
FACTORISE
To factorise you need to find two numbers that multiply
to give the constant and add to give the coefficient of x.

4. 1 Factorise x2 + 7x + 10.
List the pairs of numbers that MULTIPLY to give 10
1 and 10
−1 and −10
2 and 5
−2 and −5
and check to see if they add to give 7.
check: (x + 2)(x + 5) = x2 + 5x + 2x + 10
= x2 + 7x + 10

5. 2 Factorise x2 + 9x + 14.
List the pairs of numbers that MULTIPLY to give 14
1 and 14
−1 and −14
2 and 7
−2 and −7
and check to see if they add to give 9.
check: (x + 2)(x + 7) = x2 + 7x + 2x + 14
= x2 + 9x + 14

6. 3 Factorise x2 − 3x − 10.
List the pairs of numbers that MULTIPLY to give −10
1 and −10
−1 and 10
2 and −5
−2 and 5
and check to see if they ADD to give −3.
check: (x + 2)(x − 5) = x2 − 5x + 2x − 10
= x2 − 3x − 10

7. 4 Factorise x2 − 8x + 15.
List the pairs of numbers that MULTIPLY to give 15
1 and 15
−1 and −15
3 and 5
−3 and −5
and check to see if they ADD to give −8.
check: (x − 3)(x − 5) = x2 − 5x − 3x + 15
= x2 − 8x + 15

8. Simplifying fractions
To simplify fractions you must first factorise and then cancel the factors.

9. 1 Simplify
factorise
divide numerator and denominator by (x + 6)

10. 2 Simplify
factorise
divide numerator and denominator by (x + 9)

11. 3 Simplify
factorise
divide numerator and denominator by (x + 4)

12. The difference of two squares
EXPAND
FACTORISE
In general x2 – a2 = (x + a) (x – a)
This result is called the difference of two squares.

13. 1 Factorise x2 − 49.
check: (x + 7)(x − 7) = x2 − 7x + 7x − 49
= x2 − 49

14. 2 Factorise 4x2 − 1.
check: (2x + 1)(2x − 1) = 4x2 − 2x + 2x − 1
= 4x2 − 1

15. 2 Factorise 16x2 − 25.
check: (4x + 5)(4x − 5) = 16x2 − 20x + 20x − 25
= 16x2 − 25

16. Factorising expressions of the form ax2 + bx + c
To factorise 2x2 + 7x + 3
first look at the x2 term, 2x2 must come from 2x × x, so use (2x + ?)(x + ?)
next look at the constant term, 3 must come from 3 × 1 or 1 × 3
try expanding (2x + 3)(x + 1) and (2x + 1)(x + 3) to find which is correct So

17. 1 Factorise
5x2 come from 5x × x, so use (5x + ?)(x + ?)
−8 comes from 1 × −8 or −1 × 8 or 2 × −4 or −2 × 4 So

18. 2 Simplify
factorise
divide numerator and denominator by (x − 2)

19. 3 Simplify
factorise
divide numerator and denominator by (3x + 1)