The document discusses different methods for factorising expressions: 1) Looking for a common factor and dividing it out of all terms 2) Using the difference of two squares formula (a2 - b2 = (a - b)(a + b)) 3) Factorising quadratic trinomials into two binomial factors by identifying the values that multiply to give the constant term and sum to give the coefficient of the linear term.

1. Factorising

2. Factorising
1) Look for a common factor

3. Factorising
1) Look for a common factor
2) (i) 2 terms
 difference of two squares

4. Factorising
1) Look for a common factor
2) (i) 2 terms
 difference of two squares
 sum/difference of two cubes

5. Factorising
1) Look for a common factor
2) (i) 2 terms
 difference of two squares
 sum/difference of two cubes
(ii) 3 terms
 quadratic trinomial

6. Factorising
1) Look for a common factor
2) (i) 2 terms
 difference of two squares
 sum/difference of two cubes
(ii) 3 terms
 quadratic trinomial
(iii) 4 terms
 grouping in pairs

7. 1. Common Factor
factorising
=
dividing by common factor

8. 1. Common Factor
e.g. (i ) ax  bx
factorising
=
dividing by common factor

9. 1. Common Factor
e.g. (i ) ax  bx  x  a  b 
factorising
=
dividing by common factor

10. 1. Common Factor
e.g. (i ) ax  bx  x  a  b 
(ii ) 5 x 2  10 x
factorising
=
dividing by common factor

11. 1. Common Factor
e.g. (i ) ax  bx  x  a  b 
(ii ) 5 x 2  10 x  5 x  x  2 
factorising
=
dividing by common factor

12. 1. Common Factor
e.g. (i ) ax  bx  x  a  b 
(ii ) 5 x 2  10 x  5 x  x  2 
(iii ) mx  nx  my  ny
factorising
=
dividing by common factor

13. 1. Common Factor
e.g. (i ) ax  bx  x  a  b 
(ii ) 5 x 2  10 x  5 x  x  2 
factorising
=
dividing by common factor
(iii ) mx  nx  my  ny  x  m  n   y  m  n 

14. 1. Common Factor
e.g. (i ) ax  bx  x  a  b 
(ii ) 5 x 2  10 x  5 x  x  2 
factorising
=
dividing by common factor
(iii ) mx  nx  my  ny  x  m  n   y  m  n 
  m  n  x  y 

15. 1. Common Factor
e.g. (i ) ax  bx  x  a  b 
(ii ) 5 x 2  10 x  5 x  x  2 
factorising
=
dividing by common factor
(iii ) mx  nx  my  ny  x  m  n   y  m  n 
  m  n  x  y 
2. Difference of Two Squares
a 2  b 2   a  b  a  b 

16. 1. Common Factor
e.g. (i ) ax  bx  x  a  b 
(ii ) 5 x 2  10 x  5 x  x  2 
factorising
=
dividing by common factor
(iii ) mx  nx  my  ny  x  m  n   y  m  n 
  m  n  x  y 
2. Difference of Two Squares
a 2  b 2   a  b  a  b 
e.g. (i ) 16x 2  1

17. 1. Common Factor
e.g. (i ) ax  bx  x  a  b 
(ii ) 5 x 2  10 x  5 x  x  2 
factorising
=
dividing by common factor
(iii ) mx  nx  my  ny  x  m  n   y  m  n 
  m  n  x  y 
2. Difference of Two Squares
a 2  b 2   a  b  a  b 
e.g. (i ) 16x 2  1 =  4 x  1 4 x  1

18. 1. Common Factor
e.g. (i ) ax  bx  x  a  b 
(ii ) 5 x 2  10 x  5 x  x  2 
factorising
=
dividing by common factor
(iii ) mx  nx  my  ny  x  m  n   y  m  n 
  m  n  x  y 
2. Difference of Two Squares
a 2  b 2   a  b  a  b 
e.g. (i ) 16x 2  1 =  4 x  1 4 x  1
(ii ) 3 y 2  75

19. 1. Common Factor
e.g. (i ) ax  bx  x  a  b 
(ii ) 5 x 2  10 x  5 x  x  2 
factorising
=
dividing by common factor
(iii ) mx  nx  my  ny  x  m  n   y  m  n 
  m  n  x  y 
2. Difference of Two Squares
a 2  b 2   a  b  a  b 
e.g. (i ) 16x 2  1 =  4 x  1 4 x  1
(ii ) 3 y 2  75 =3  y 2  25 

20. 1. Common Factor
e.g. (i ) ax  bx  x  a  b 
(ii ) 5 x 2  10 x  5 x  x  2 
factorising
=
dividing by common factor
(iii ) mx  nx  my  ny  x  m  n   y  m  n 
  m  n  x  y 
2. Difference of Two Squares
a 2  b 2   a  b  a  b 
e.g. (i ) 16x 2  1 =  4 x  1 4 x  1
(ii ) 3 y 2  75 =3  y 2  25 
=3  y  5  y  5 

21. 1. Common Factor
e.g. (i ) ax  bx  x  a  b 
(ii ) 5 x 2  10 x  5 x  x  2 
factorising
=
dividing by common factor
(iii ) mx  nx  my  ny  x  m  n   y  m  n 
  m  n  x  y 
2. Difference of Two Squares
a 2  b 2   a  b  a  b 
e.g. (i ) 16x 2  1 =  4 x  1 4 x  1
(ii ) 3 y 2  75 =3  y 2  25 
=3  y  5  y  5 
(iii ) 5 x  5 y  x 2  y 2

22. 1. Common Factor
e.g. (i ) ax  bx  x  a  b 
(ii ) 5 x 2  10 x  5 x  x  2 
factorising
=
dividing by common factor
(iii ) mx  nx  my  ny  x  m  n   y  m  n 
  m  n  x  y 
2. Difference of Two Squares
a 2  b 2   a  b  a  b 
e.g. (i ) 16x 2  1 =  4 x  1 4 x  1
(ii ) 3 y 2  75 =3  y 2  25 
=3  y  5  y  5 
(iii ) 5 x  5 y  x 2  y 2 =5  x  y    x  y  x  y 

23. 1. Common Factor
e.g. (i ) ax  bx  x  a  b 
(ii ) 5 x 2  10 x  5 x  x  2 
factorising
=
dividing by common factor
(iii ) mx  nx  my  ny  x  m  n   y  m  n 
  m  n  x  y 
2. Difference of Two Squares
a 2  b 2   a  b  a  b 
e.g. (i ) 16x 2  1 =  4 x  1 4 x  1
(ii ) 3 y 2  75 =3  y 2  25 
=3  y  5  y  5 
(iii ) 5 x  5 y  x 2  y 2 =5  x  y    x  y  x  y 
=  x  y  5  x  y 

24. 3. Quadratic Trinomial
a) Monic Quadratic
 x  a  x  b   x 2   a  b  x  ab

25. 3. Quadratic Trinomial
a) Monic Quadratic
 x  a  x  b   x 2   a  b  x  ab
e.g. (i ) x 2  9 x  18

26. 3. Quadratic Trinomial
a) Monic Quadratic
 x  a  x  b   x 2   a  b  x  ab
e.g. (i ) x 2  9 x  18
  18
9

27. 3. Quadratic Trinomial
a) Monic Quadratic
 x  a  x  b   x 2   a  b  x  ab
e.g. (i ) x 2  9 x  18
  x  6  x  3
  18
9

28. 3. Quadratic Trinomial
a) Monic Quadratic
 x  a  x  b   x 2   a  b  x  ab
e.g. (i ) x 2  9 x  18
  x  6  x  3
(ii ) t 2  4t  3
  18
9

29. 3. Quadratic Trinomial
a) Monic Quadratic
 x  a  x  b   x 2   a  b  x  ab
e.g. (i ) x 2  9 x  18
  x  6  x  3
(ii ) t 2  4t  3
  18
9
3
  4

30. 3. Quadratic Trinomial
a) Monic Quadratic
 x  a  x  b   x 2   a  b  x  ab
e.g. (i ) x 2  9 x  18
  x  6  x  3
(ii ) t 2  4t  3
  t  3 t  1
  18
9
3
  4

31. 3. Quadratic Trinomial
a) Monic Quadratic
 x  a  x  b   x 2   a  b  x  ab
e.g. (i ) x 2  9 x  18
  x  6  x  3
(ii ) t 2  4t  3
  t  3 t  1
(iii ) x 2  5 xy  4 y 2
  18
9
3
  4

32. 3. Quadratic Trinomial
a) Monic Quadratic
 x  a  x  b   x 2   a  b  x  ab
e.g. (i ) x 2  9 x  18
  x  6  x  3
(ii ) t 2  4t  3
  t  3 t  1
  18
9
3
  4
(iii ) x 2  5 xy  4 y 2   4 y 2
  5 y

33. 3. Quadratic Trinomial
a) Monic Quadratic
 x  a  x  b   x 2   a  b  x  ab
e.g. (i ) x 2  9 x  18
  x  6  x  3
(ii ) t 2  4t  3
  t  3 t  1
  18
9
3
  4
(iii ) x 2  5 xy  4 y 2   4 y 2
  x  y  x  4 y    5 y

34. b) Splitting the Middle

35. b) Splitting the Middle
e.g. (i ) 3x 2  4 x  7

36. b) Splitting the Middle
e.g. (i ) 3x  4 x  7
2
  21
4
Multiply the constant by the
coefficient of x squared
7  3

37. b) Splitting the Middle
  21
e.g. (i ) 3x  4 x  7
 3x 2  3x  7 x  7   4
2
Multiply the constant by the
coefficient of x squared
7  3

38. b) Splitting the Middle
  21
e.g. (i ) 3x  4 x  7
 3x 2  3x  7 x  7   4
 3 x  x  1  7  x  1
2
Multiply the constant by the
coefficient of x squared
7  3

39. b) Splitting the Middle
  21
e.g. (i ) 3x  4 x  7
 3x 2  3x  7 x  7   4
 3 x  x  1  7  x  1
2
  x  1 3 x  7 
Multiply the constant by the
coefficient of x squared
7  3

40. b) Splitting the Middle
  21
e.g. (i ) 3x  4 x  7
 3x 2  3x  7 x  7   4
 3 x  x  1  7  x  1
2
  x  1 3 x  7 
(ii ) 2 x 2  5 x  12
Multiply the constant by the
coefficient of x squared
7  3

41. b) Splitting the Middle
  21
e.g. (i ) 3x  4 x  7
 3x 2  3x  7 x  7   4
 3 x  x  1  7  x  1
2
  x  1 3 x  7 
(ii ) 2 x 2  5 x  12
  24
  5
Multiply the constant by the
coefficient of x squared
7  3

42. b) Splitting the Middle
  21
e.g. (i ) 3x  4 x  7
 3x 2  3x  7 x  7   4
 3 x  x  1  7  x  1
2
  x  1 3 x  7 
  24
(ii ) 2 x 2  5 x  12
 2 x 2  8 x  3 x  12   5
Multiply the constant by the
coefficient of x squared
7  3

43. b) Splitting the Middle
  21
e.g. (i ) 3x  4 x  7
 3x 2  3x  7 x  7   4
 3 x  x  1  7  x  1
2
  x  1 3 x  7 
  24
(ii ) 2 x 2  5 x  12
 2 x 2  8 x  3 x  12   5
 2 x  x  4  3 x  4
Multiply the constant by the
coefficient of x squared
7  3

44. b) Splitting the Middle
  21
e.g. (i ) 3x  4 x  7
 3x 2  3x  7 x  7   4
 3 x  x  1  7  x  1
2
  x  1 3 x  7 
  24
(ii ) 2 x 2  5 x  12
 2 x 2  8 x  3 x  12   5
 2 x  x  4  3 x  4
  x  4  2 x  3
Multiply the constant by the
coefficient of x squared
7  3

45. b) Splitting the Middle
  21
e.g. (i ) 3x  4 x  7
 3x 2  3x  7 x  7   4
 3 x  x  1  7  x  1
2
Multiply the constant by the
coefficient of x squared
7  3
  x  1 3 x  7 
  24
(ii ) 2 x 2  5 x  12
 2 x 2  8 x  3 x  12   5
 2 x  x  4  3 x  4
  x  4  2 x  3
Exercise 1C; 1e, 2f, 3d, 4ejo, 5adhkn, 6ace etc, 7ace etc, 8*bdfij