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;
• factorised as a mixture of quadratic and linear factors over the
real field
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
• factorised to n linear factors over the complex 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
NOTE: odd ordered polynomials must have a real root
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
e.g . i x 2 2 x 2
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 x 12 1
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
x 1 i x 1 i
11. ii z 4 z 2 12 0
z 3z 4 0
z 3 z 3z 4 0
2
2
2
factorised over Real numbers
12. ii z 4 z 2 12 0
z 3z 4 0
factorised over Real numbers
z 3 z 3z 4 0
z 3 z 3 z 2i z 2i 0 factorised over Complex numbers
2
2
2
13. ii z 4 z 2 12 0
z 3z 4 0
factorised over Real numbers
z 3 z 3z 4 0
z 3 z 3 z 2i z 2i 0 factorised over Complex numbers
2
2
2
z 3 or z 2i
14. ii z 4 z 2 12 0
z 3z 4 0
factorised over Real numbers
z 3 z 3z 4 0
z 3 z 3 z 2i z 2i 0 factorised over Complex numbers
2
2
2
z 3 or z 2i
If (x – a) is a factor of P(x), then P(a) = 0
b
If (ax – b) is a factor of P(x), then P = 0
a
15. ii z 4 z 2 12 0
z 3z 4 0
factorised over Real numbers
z 3 z 3z 4 0
z 3 z 3 z 2i z 2i 0 factorised over Complex numbers
2
2
2
z 3 or z 2i
If (x – a) is a factor of P(x), then P(a) = 0
b
If (ax – b) is a factor of P(x), then P = 0
a
iii Factorise 2 x 3 3x 2 8 x 5
16. ii z 4 z 2 12 0
z 3z 4 0
factorised over Real numbers
z 3 z 3z 4 0
z 3 z 3 z 2i z 2i 0 factorised over Complex numbers
2
2
2
z 3 or z 2i
If (x – a) is a factor of P(x), then P(a) = 0
b
If (ax – b) is a factor of P(x), then P = 0
a
iii Factorise 2 x 3 3x 2 8 x 5
as it is a cubic it must have a real factor
17. ii z 4 z 2 12 0
z 3z 4 0
factorised over Real numbers
z 3 z 3z 4 0
z 3 z 3 z 2i z 2i 0 factorised over Complex numbers
2
2
2
z 3 or z 2i
If (x – a) is a factor of P(x), then P(a) = 0
b
If (ax – b) is a factor of P(x), then P = 0
a
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 5
P 2 3 8
2 2
2
2
1 3
45
4 4
0
18. ii z 4 z 2 12 0
z 3z 4 0
factorised over Real numbers
z 3 z 3z 4 0
z 3 z 3 z 2i z 2i 0 factorised over Complex numbers
2
2
2
z 3 or z 2i
If (x – a) is a factor of P(x), then P(a) = 0
b
If (ax – b) is a factor of P(x), then P = 0
a
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 5
2 x3 3x 2 8 x 5
P 2 3 8
2 2
2
2
2 x 1 x 2 2 x 5
1 3
45
4 4
0
19. ii z 4 z 2 12 0
z 3z 4 0
factorised over Real numbers
z 3 z 3z 4 0
z 3 z 3 z 2i z 2i 0 factorised over Complex numbers
2
2
2
z 3 or z 2i
If (x – a) is a factor of P(x), then P(a) = 0
b
If (ax – b) is a factor of P(x), then P = 0
a
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 5
2 x3 3x 2 8 x 5
P 2 3 8
2 2
2
2
2 x 1 x 2 2 x 5
1 3
45
2
2 x 1 x 1 4
4 4
0
20. ii z 4 z 2 12 0
z 3z 4 0
factorised over Real numbers
z 3 z 3z 4 0
z 3 z 3 z 2i z 2i 0 factorised over Complex numbers
2
2
2
z 3 or z 2i
If (x – a) is a factor of P(x), then P(a) = 0
b
If (ax – b) is a factor of P(x), then P = 0
a
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 5
2 x3 3x 2 8 x 5
P 2 3 8
2 2
2
2
2 x 1 x 2 2 x 5
1 3
45
2
2 x 1 x 1 4
4 4
2 x 1 x 1 2i x 1 2i
0
21. (iv) Given that P x 4 x 4 8 x 3 5 x 2 x 3 has two rational zeros,
find these zeros and factorise P(x) over the complex field.
22. (iv) Given that P x 4 x 4 8 x 3 5 x 2 x 3 has two rational zeros,
find these zeros and factorise P(x) over the complex field.
1 4 1 8 1 5 1 1 3
P
2 16 8 4 2
0
2 x 1 is a factor
23. (iv) Given that P x 4 x 4 8 x 3 5 x 2 x 3 has two rational zeros,
find these zeros and factorise P(x) over the complex field.
1 4 1 8 1 5 1 1 3
P
2 16 8 4 2
0
2 x 1 is a factor
P x 2 x 12 x 3
3
24. (iv) Given that P x 4 x 4 8 x 3 5 x 2 x 3 has two rational zeros,
find these zeros and factorise P(x) over the complex field.
1 4 1 8 1 5 1 1 3
P
2 16 8 4 2
0
2 x 1 is a factor
P x 2 x 12 x 3
5 x 3
25. (iv) Given that P x 4 x 4 8 x 3 5 x 2 x 3 has two rational zeros,
find these zeros and factorise P(x) over the complex field.
1 4 1 8 1 5 1 1 3
P
2 16 8 4 2
0
2 x 1 is a factor
P x 2 x 12 x 3 5x 2 5 x 3
26. (iv) Given that P x 4 x 4 8 x 3 5 x 2 x 3 has two rational zeros,
find these zeros and factorise P(x) over the complex field.
1 4 1 8 1 5 1 1 3
P
2 16 8 4 2
0
2 x 1 is a factor
P x 2 x 12 x 3 5x 2 5 x 3
3 27 9 3
P 2
5 5 3
2 8 4 2
0
27. (iv) Given that P x 4 x 4 8 x 3 5 x 2 x 3 has two rational zeros,
find these zeros and factorise P(x) over the complex field.
1 4 1 8 1 5 1 1 3
P
2 16 8 4 2
0
2 x 1 is a factor
P x 2 x 12 x 3 5x 2 5 x 3
3 27 9 3
P 2
5 5 3
2 8 4 2
0
2 x 3 is a factor
3
1
rational zeros are and 2
2
29. P x 4 x 4 8 x 3 5 x 2 x 3
2 x 1 2 x 3 x 2
1
30. P x 4 x 4 8 x 3 5 x 2 x 3
2 x 1 2 x 3 x 2
1
1 3 2 x 6 x
1 2 x 1 2 x
31. P x 4 x 4 8 x 3 5 x 2 x 3
2 x 1 2 x 3 x 2
1
1 3 2 x 6 x
1 2 x 1 2 x
4x
1 3 ? x 3 x
32. P x 4 x 4 8 x 3 5 x 2 x 3
2 x 1 2 x 3 x 2 x
1
1 3 2 x 6 x
1 2 x 1 2 x
4x
1 3 ? x 3 x
33. P x 4 x 4 8 x 3 5 x 2 x 3
2 x 1 2 x 3 x 2 x
1
2
1 3
2 x 12 x 3 x
2 4
1 3 2 x 6 x
1 2 x 1 2 x
4x
1 3 ? x 3 x
34. P x 4 x 4 8 x 3 5 x 2 x 3
1 3 2 x 6 x
2 x 1 2 x 3 x 2 x
1
1 2 x 1 2 x
2
1 3
4x
2 x 12 x 3 x
2 4
1 3 ? x 3 x
1
3
1
3
2 x 12 x 3 x
i x
i
2 2
2 2
35. P x 4 x 4 8 x 3 5 x 2 x 3
1 3 2 x 6 x
2 x 1 2 x 3 x 2 x
1
1 2 x 1 2 x
2
1 3
4x
2 x 12 x 3 x
2 4
1 3 ? x 3 x
1
3
1
3
2 x 12 x 3 x
i x
i
2 2
2 2
Cambridge: Exercise 5A; 1b, 2, 3, 5, 6, 7b, 8, 9, 10, 12 to 15