2. Factorising Complex
Expressions
If a polynomial’s coefficients are all real then the roots will appear
in complex conjugate pairs.
3. Factorising Complex
Expressions
If a polynomial’s coefficients are all real then the roots will appear
in complex conjugate pairs.
Every polynomial of degree n can be;
4. Factorising Complex
Expressions
If a polynomial’s coefficients are all real then the roots will appear
in complex conjugate pairs.
Every polynomial of degree n can be;
• factorised as a mixture of quadratic and linear factors over the
real field
5. Factorising Complex
Expressions
If a polynomial’s coefficients are all real then the roots will appear
in complex conjugate pairs.
Every polynomial of degree n can be;
• factorised as a mixture of quadratic and linear factors over the
real field
• factorised to n linear factors over the complex field
6. Factorising Complex
Expressions
If a polynomial’s coefficients are all real then the roots will appear
in complex conjugate pairs.
Every polynomial of degree n can be;
• factorised as a mixture of quadratic and linear factors over the
real field
• factorised to n linear factors over the complex field
NOTE: odd ordered polynomials must have a real root
7. Factorising Complex
Expressions
If a polynomial’s coefficients are all real then the roots will appear
in complex conjugate pairs.
Every polynomial of degree n can be;
• factorised as a mixture of quadratic and linear factors over the
real field
• factorised to n linear factors over the complex field
NOTE: odd ordered polynomials must have a real root
e.g . i x 2 2 x 2
8. Factorising Complex
Expressions
If a polynomial’s coefficients are all real then the roots will appear
in complex conjugate pairs.
Every polynomial of degree n can be;
• factorised as a mixture of quadratic and linear factors over the
real field
• factorised to n linear factors over the complex field
NOTE: odd ordered polynomials must have a real root
e.g . i x 2 2 x 2 x 12 1
9. Factorising Complex
Expressions
If a polynomial’s coefficients are all real then the roots will appear
in complex conjugate pairs.
Every polynomial of degree n can be;
• factorised as a mixture of quadratic and linear factors over the
real field
• factorised to n linear factors over the complex field
NOTE: odd ordered polynomials must have a real root
e.g . i x 2 2 x 2 x 12 1
x 1 i x 1 i
13. ii z 4 z 2 12 0
z 3z 4 0
2 2
z 3 z 3z 4 0
2
factorised over Real numbers
14. ii z 4 z 2 12 0
z 3z 4 0
2 2
z 3 z 3z 4 0
2
factorised over Real numbers
z 3 z 3 z 2i z 2i 0
15. ii z 4 z 2 12 0
z 3z 4 0
2 2
z 3 z 3z 4 0
2
factorised over Real numbers
z 3 z 3 z 2i z 2i 0 factorised over Complex numbers
16. ii z 4 z 2 12 0
z 3z 4 0
2 2
z 3 z 3z 4 0
2
factorised over Real numbers
z 3 z 3 z 2i z 2i 0 factorised over Complex numbers
z 3 or z 2i
17. ii z 4 z 2 12 0
z 3z 4 0
2 2
z 3 z 3z 4 0
2
factorised over Real numbers
z 3 z 3 z 2i z 2i 0 factorised over Complex numbers
z 3 or z 2i
iii Factorise 2 x 3 3x 2 8 x 5
18. ii z 4 z 2 12 0
z 3z 4 0
2 2
z 3 z 3z 4 0
2
factorised over Real numbers
z 3 z 3 z 2i z 2i 0 factorised over Complex numbers
z 3 or z 2i
iii Factorise 2 x 3 3x 2 8 x 5
as it is a cubic it must have a real factor
19. ii z 4 z 2 12 0
z 3z 4 0
2 2
z 3 z 3z 4 0
2
factorised over Real numbers
z 3 z 3 z 2i z 2i 0 factorised over Complex numbers
z 3 or z 2i
iii Factorise 2 x 3 3x 2 8 x 5
as it is a cubic it must have a real factor
3 2
1 1 1 1
P 2 3 8 5
2 2 2 2
1 3
45
4 4
0
20. ii z 4 z 2 12 0
z 3z 4 0
2 2
z 3 z 3z 4 0
2
factorised over Real numbers
z 3 z 3 z 2i z 2i 0 factorised over Complex numbers
z 3 or z 2i
iii Factorise 2 x 3 3x 2 8 x 5
as it is a cubic it must have a real factor
3 2
1 1 1 1
P 2 3 8 5
2 2 2 2
1 3
45
4 4
0
2 x 3 3 x 2 8 x 5 2 x 1x 2 2 x 5
21. ii z 4 z 2 12 0
z 3z 4 0
2 2
z 3 z 3z 4 0
2
factorised over Real numbers
z 3 z 3 z 2i z 2i 0 factorised over Complex numbers
z 3 or z 2i
iii Factorise 2 x 3 3x 2 8 x 5
as it is a cubic it must have a real factor
3 2
1 1 1 1
P 2 3 8 5
2 2 2 2
1 3
45
4 4
0
2 x 3 3 x 2 8 x 5 2 x 1x 2 2 x 5
2 x 1 x 1 4
2
22. ii z 4 z 2 12 0
z 3z 4 0
2 2
z 3 z 3z 4 0
2
factorised over Real numbers
z 3 z 3 z 2i z 2i 0 factorised over Complex numbers
z 3 or z 2i
iii Factorise 2 x 3 3x 2 8 x 5
as it is a cubic it must have a real factor
3 2
1 1 1 1
P 2 3 8 5
2 2 2 2
1 3
45
4 4
0
2 x 3 3 x 2 8 x 5 2 x 1x 2 2 x 5
2 x 1 x 1 4
2
2 x 1 x 1 2i x 1 2i
24. iv z 5 z 4 z 3 z 2 z 1 0
Now z 6 1 z 1z 5 z 4 z 3 z 2 z 1
25. iv z 5 z 4 z 3 z 2 z 1 0
Now z 6 1 z 1z 5 z 4 z 3 z 2 z 1
And z 6 1 0 has solutions;
2 k
z cis k 0, 1, 2,3
6
26. iv z 5 z 4 z 3 z 2 z 1 0
Now z 6 1 z 1z 5 z 4 z 3 z 2 z 1
And z 6 1 0 has solutions;
2 k
z cis k 0, 1, 2,3
6
z5 z4 z3 z2 1 0
27. iv z 5 z 4 z 3 z 2 z 1 0
Now z 6 1 z 1z 5 z 4 z 3 z 2 z 1
And z 6 1 0 has solutions;
2 k
z cis k 0, 1, 2,3
6
z5 z4 z3 z2 1 0
z 1z 5 z 4 z 3 z 2 1
0
z 1
28. iv z 5 z 4 z 3 z 2 z 1 0
Now z 6 1 z 1z 5 z 4 z 3 z 2 z 1
And z 6 1 0 has solutions;
2 k
z cis k 0, 1, 2,3
6
z5 z4 z3 z2 1 0
z 1z 5 z 4 z 3 z 2 1
0
z 1
z 6 1 0, z 1
29. iv z 5 z 4 z 3 z 2 z 1 0
Now z 6 1 z 1z 5 z 4 z 3 z 2 z 1
And z 6 1 0 has solutions;
2 k
z cis k 0, 1, 2,3
6
z5 z4 z3 z2 1 0
z 1z 5 z 4 z 3 z 2 1
0
z 1
z 6 1 0, z 1
2 2
z cis , cis , cis , cis , cis
3 3 3 3
30. iv z 5 z 4 z 3 z 2 z 1 0
Now z 6 1 z 1z 5 z 4 z 3 z 2 z 1
And z 6 1 0 has solutions;
2 k
z cis k 0, 1, 2,3
6
z5 z4 z3 z2 1 0
z 1z 5 z 4 z 3 z 2 1
0
z 1
z 6 1 0, z 1
2 2
z cis , cis , cis , cis , cis
3 3 3 3
1 3 1 3 1 3 1 3
z i, i, i, i, 1
2 2 2 2 2 2 2 2
31. v 1996 HSC
2 2
Let cos i sin
9 9
a) Show that k is a solution of z 9 1 0, where k is an integer
32. v 1996 HSC
2 2
Let cos i sin
9 9
a) Show that k is a solution of z 9 1 0, where k is an integer
z9 1
33. v 1996 HSC
2 2
Let cos i sin
9 9
a) Show that k is a solution of z 9 1 0, where k is an integer
z9 1
2k
z cis
9 k 0,1,2,3,4,5,6,7,8
34. v 1996 HSC
2 2
Let cos i sin
9 9
a) Show that k is a solution of z 9 1 0, where k is an integer
z9 1
2k
z cis
9 k 0,1,2,3,4,5,6,7,8
2
k
z cis
9
35. v 1996 HSC
2 2
Let cos i sin
9 9
a) Show that k is a solution of z 9 1 0, where k is an integer
z9 1
2k
z cis
9 k 0,1,2,3,4,5,6,7,8
2
k
z cis
9
z k
36. v 1996 HSC
2 2
Let cos i sin
9 9
a) Show that k is a solution of z 9 1 0, where k is an integer
z9 1
2k
z cis
9 k 0,1,2,3,4,5,6,7,8
2
k
z cis
9
z k
b) Prove that 2 3 4 5 6 7 8 1
37. v 1996 HSC
2 2
Let cos i sin
9 9
a) Show that k is a solution of z 9 1 0, where k is an integer
z9 1
2k
z cis
9 k 0,1,2,3,4,5,6,7,8
2
k
z cis
9
z k
b) Prove that 2 3 4 5 6 7 8 1
z9 1 0
38. v 1996 HSC
2 2
Let cos i sin
9 9
a) Show that k is a solution of z 9 1 0, where k is an integer
z9 1
2k
z cis
9 k 0,1,2,3,4,5,6,7,8
2
k
z cis
9
z k
b) Prove that 2 3 4 5 6 7 8 1
z9 1 0
1 2 3 4 5 6 7 8 0
39. v 1996 HSC
2 2
Let cos i sin
9 9
a) Show that k is a solution of z 9 1 0, where k is an integer
z9 1
2k
z cis
9 k 0,1,2,3,4,5,6,7,8
2
k
z cis
9
z k
b) Prove that 2 3 4 5 6 7 8 1
z9 1 0
1 2 3 4 5 6 7 8 0 sum of roots
40. v 1996 HSC
2 2
Let cos i sin
9 9
a) Show that k is a solution of z 9 1 0, where k is an integer
z9 1
2k
z cis
9 k 0,1,2,3,4,5,6,7,8
2
k
z cis
9
z k
b) Prove that 2 3 4 5 6 7 8 1
z9 1 0
1 2 3 4 5 6 7 8 0 sum of roots
2 3 4 5 6 7 8 1
41. 2 4 1
c) Hence show that cos cos cos
9 9 9 8
42. 2 4 1
c) Hence show that cos cos cos
9 9 9 8
A B A B
sin A sin B 2sin cos
2 2
43. 2 4 1
c) Hence show that cos cos cos
9 9 9 8
A B A B (2 sine half sum
sin A sin B 2sin cos
2 2 cos half diff)
44. 2 4 1
c) Hence show that cos cos cos
9 9 9 8
A B A B (2 sine half sum
sin A sin B 2sin cos
2 2 cos half diff)
A B cos A B
cos A cos B 2 cos
2 2
45. 2 4 1
c) Hence show that cos cos cos
9 9 9 8
A B A B (2 sine half sum
sin A sin B 2sin cos
2 2 cos half diff)
A B cos A B
cos A cos B 2 cos
(2 cos half sum
2 2 cos half diff)
46. 2 4 1
c) Hence show that cos cos cos
9 9 9 8
A B A B (2 sine half sum
sin A sin B 2sin cos
2 2 cos half diff)
A B cos A B
cos A cos B 2 cos
(2 cos half sum
2 2 cos half diff)
A B A B
cos A cos B 2sin sin
2 2
47. 2 4 1
c) Hence show that cos cos cos
9 9 9 8
A B A B (2 sine half sum
sin A sin B 2sin cos
2 2 cos half diff)
A B cos A B
cos A cos B 2 cos
(2 cos half sum
2 2 cos half diff)
A B A B (minus 2 sine half sum
cos A cos B 2sin sin
2 2 sine half diff)
48. 2 4 1
c) Hence show that cos cos cos
9 9 9 8
A B A B (2 sine half sum
sin A sin B 2sin cos
2 2 cos half diff)
A B cos A B
cos A cos B 2 cos
(2 cos half sum
2 2 cos half diff)
A B A B (minus 2 sine half sum
cos A cos B 2sin sin
2 2 sine half diff)
2 3 4 5 6 7 8 1
49. 2 4 1
c) Hence show that cos cos cos
9 9 9 8
A B A B (2 sine half sum
sin A sin B 2sin cos
2 2 cos half diff)
A B cos A B
cos A cos B 2 cos
(2 cos half sum
2 2 cos half diff)
A B A B (minus 2 sine half sum
cos A cos B 2sin sin
2 2 sine half diff)
2 3 4 5 6 7 8 1
roots appear in conjugate pairs
50. 2 4 1
c) Hence show that cos cos cos
9 9 9 8
A B A B (2 sine half sum
sin A sin B 2sin cos
2 2 cos half diff)
A B cos A B
cos A cos B 2 cos
(2 cos half sum
2 2 cos half diff)
A B A B (minus 2 sine half sum
cos A cos B 2sin sin
2 2 sine half diff)
2 3 4 5 6 7 8 1
roots appear in conjugate pairs
2 4 6 8
2cos 2cos 2cos 2cos 1
9 9 9 9
51. 2 4 1
c) Hence show that cos cos cos
9 9 9 8
A B A B (2 sine half sum
sin A sin B 2sin cos
2 2 cos half diff)
A B cos A B
cos A cos B 2 cos
(2 cos half sum
2 2 cos half diff)
A B A B (minus 2 sine half sum
cos A cos B 2sin sin
2 2 sine half diff)
2 3 4 5 6 7 8 1
roots appear in conjugate pairs
2 4 6 8
2cos 2cos 2cos 2cos 1
9 9 9 9
2 4 6 8 1
cos cos cos cos
9 9 9 9 2
52. 2 4 1
c) Hence show that cos cos cos
9 9 9 8
A B A B (2 sine half sum
sin A sin B 2sin cos
2 2 cos half diff)
A B cos A B
cos A cos B 2 cos
(2 cos half sum
2 2 cos half diff)
A B A B (minus 2 sine half sum
cos A cos B 2sin sin
2 2 sine half diff)
2 3 4 5 6 7 8 1
roots appear in conjugate pairs
2 4 6 8
2cos 2cos 2cos 2cos 1
9 9 9 9
2 4 6 8 1
cos cos cos cos
9 9 9 9 2
3 7 1
2cos cos 2cos cos
9 9 9 9 2
54. 3 7 1
2cos cos 2cos cos
9 9 9 9 2
3 7 1
cos cos cos
9 9 9 4
55. 3 7 1
2cos cos 2cos cos
9 9 9 9 2
3 7 1
cos cos cos
9 9 9 4
5 2 1
cos 2cos cos
9 9 9 4
56. 3 7 1
2cos cos 2cos cos
9 9 9 9 2
3 7 1
cos cos cos
9 9 9 4
5 2 1
cos 2cos cos
9 9 9 4
2 5 1
cos cos cos
9 9 9 8
57. 3 7 1
2cos cos 2cos cos
9 9 9 9 2
3 7 1
cos cos cos
9 9 9 4
5 2 1
cos 2cos cos
9 9 9 4
2 5 1
cos cos cos
9 9 9 8
5 4
But cos cos
9 9
58. 3 7 1
2cos cos 2cos cos
9 9 9 9 2
3 7 1
cos cos cos
9 9 9 4
5 2 1
cos 2cos cos
9 9 9 4
2 5 1
cos cos cos
9 9 9 8
5 4
But cos cos
9 9
2 4 1
cos cos cos
9 9 9 8
59. 3 7 1
2cos cos 2cos cos
9 9 9 9 2
3 7 1
cos cos cos
9 9 9 4
5 2 1
cos 2cos cos
9 9 9 4
2 5 1
cos cos cos
9 9 9 8
5 4
But cos cos
9 9
2 4 1
cos cos cos
9 9 9 8
2 4 1
cos cos cos
9 9 9 8
60. OR
z9 1
z 1 z z 8 z 2 z 7 z 3 z 6 z 4 z 5
61. OR
z9 1
z 1 z z 8 z 2 z 7 z 3 z 6 z 4 z 5
2 4
z 1 z 2 2cos z 1 z 2 2cos z 1
9 9
2 6 2 8
z 2cos z 1 z 2cos z 1
9 9
62. OR
z9 1
z 1 z z 8 z 2 z 7 z 3 z 6 z 4 z 5
2 4
z 1 z 2 2cos z 1 z 2 2cos z 1
9 9
2 6 2 8
z 2cos z 1 z 2cos z 1
9 9
2 2 2 4
z 1 z 2cos z 1 z 2cos z 1
9 9
8
z 2 z 1 z 2 2cos
9
z 1
63. OR
z9 1
z 1 z z 8 z 2 z 7 z 3 z 6 z 4 z 5
2 4
z 1 z 2 2cos z 1 z 2 2cos z 1
9 9
2 6 2 8
z 2cos z 1 z 2cos z 1
9 9
2 2 2 4
z 1 z 2cos z 1 z 2cos z 1
9 9
8
z 2 z 1 z 2 2cos
9
z 1
Let z i
64. OR
z9 1
z 1 z z 8 z 2 z 7 z 3 z 6 z 4 z 5
2 4
z 1 z 2 2cos z 1 z 2 2cos z 1
9 9
2 6 2 8
z 2cos z 1 z 2cos z 1
9 9
2 2 2 4
z 1 z 2cos z 1 z 2cos z 1
9 9
8
z 2 z 1 z 2 2cos
9
z 1
Let z i
2 4 8
i 9 1 i 1 2cos i 2cos i i 2cos i
9 9 9
65. 2 4 8
i 1 i 1 2cos
9
i 2cos i i 2cos i
9 9 9
66. 2 4 8
i 1 i 1 2cos
9
i 2cos i i 2cos i
9 9 9
2 4 8
i 1 i i 1 2cos
4
2cos 2cos
9 9 9
67. 2 4 8
i 1 i 1 2cos
9
i 2cos i i 2cos i
9 9 9
2 4 8
i 1 i i 1 2cos
4
2cos 2cos
9 9 9
2 4 8
1 8cos cos cos
9 9 9
68. 2 4 8
i 1 i 1 2cos
9
i 2cos i i 2cos i
9 9 9
2 4 8
i 1 i i 1 2cos
4
2cos 2cos
9 9 9
2 4 8
1 8cos cos cos
9 9 9
2 4
1 8cos cos cos
9 9 9
69. 2 4 8
i 1 i 1 2cos
9
i 2cos i i 2cos i
9 9 9
2 4 8
i 1 i i 1 2cos
4
2cos 2cos
9 9 9
2 4 8
1 8cos cos cos
9 9 9
2 4
1 8cos cos cos
9 9 9
2 4 1
cos cos cos
9 9 9 8
70. 2 4 8
i 1 i 1 2cos
9
i 2cos i i 2cos i
9 9 9
2 4 8
i 1 i i 1 2cos
4
2cos 2cos
9 9 9
2 4 8
1 8cos cos cos
9 9 9
2 4
1 8cos cos cos
9 9 9
2 4 1
cos cos cos
9 9 9 8
Exercise 4J; 1 to 4, 7ac