x2 t01 08 factorising complex expressions (2010)
TRANSCRIPT
Factorising Complex Expressions
Factorising Complex Expressions
If a polynomial’s coefficients are all real then the roots will appear in complex conjugate pairs.
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;
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
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
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 fieldNOTE: odd ordered polynomials must have a real root
Factorising Complex Expressions
22 .. 2 xxige
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 fieldNOTE: odd ordered polynomials must have a real root
Factorising Complex Expressions
22 .. 2 xxige 11 2 x
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 fieldNOTE: odd ordered polynomials must have a real root
Factorising Complex Expressions
22 .. 2 xxige 11 2 x
If a polynomial’s coefficients are all real then the roots will appear in complex conjugate pairs.
ixix 11
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 fieldNOTE: odd ordered polynomials must have a real root
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81
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92cos
9cos that show Hence c)
81
94cos
92cos
9cos that show Hence c)
sin sin 2sin cos2 2
A B A BA B
81
94cos
92cos
9cos that show Hence c)
sin sin 2sin cos2 2
A B A BA B
(2 sine half sum cos half diff)
81
94cos
92cos
9cos that show Hence c)
sin sin 2sin cos2 2
A B A BA B
2
cos2
cos2coscos BABABA
(2 sine half sum cos half diff)
81
94cos
92cos
9cos that show Hence c)
sin sin 2sin cos2 2
A B A BA B
2
cos2
cos2coscos BABABA
(2 sine half sum cos half diff)
(2 cos half sum cos half diff)
81
94cos
92cos
9cos that show Hence c)
sin sin 2sin cos2 2
A B A BA B
2
cos2
cos2coscos BABABA
cos cos 2sin sin2 2
A B A BA B
(2 sine half sum cos half diff)
(2 cos half sum cos half diff)
81
94cos
92cos
9cos that show Hence c)
sin sin 2sin cos2 2
A B A BA B
2
cos2
cos2coscos BABABA
cos cos 2sin sin2 2
A B A BA B
(2 sine half sum cos half diff)
(2 cos half sum cos half diff)
(minus 2 sine half sum sine half diff)
81
94cos
92cos
9cos that show Hence c)
sin sin 2sin cos2 2
A B A BA B
2
cos2
cos2coscos BABABA
cos cos 2sin sin2 2
A B A BA B
18765432
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(2 cos half sum cos half diff)
(minus 2 sine half sum sine half diff)
81
94cos
92cos
9cos that show Hence c)
sin sin 2sin cos2 2
A B A BA B
2
cos2
cos2coscos BABABA
cos cos 2sin sin2 2
A B A BA B
18765432 roots appear in conjugate pairs
(2 sine half sum cos half diff)
(2 cos half sum cos half diff)
(minus 2 sine half sum sine half diff)
81
94cos
92cos
9cos that show Hence c)
sin sin 2sin cos2 2
A B A BA B
2
cos2
cos2coscos BABABA
cos cos 2sin sin2 2
A B A BA B
18765432 roots appear in conjugate pairs
2 4 6 82cos 2cos 2cos 2cos 19 9 9 9
(2 sine half sum cos half diff)
(2 cos half sum cos half diff)
(minus 2 sine half sum sine half diff)
81
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92cos
9cos that show Hence c)
sin sin 2sin cos2 2
A B A BA B
2
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cos cos 2sin sin2 2
A B A BA B
18765432 roots appear in conjugate pairs
2 4 6 82cos 2cos 2cos 2cos 19 9 9 9
(2 sine half sum cos half diff)
(2 cos half sum cos half diff)
(minus 2 sine half sum sine half diff)
2 4 6 8 1cos cos cos cos9 9 9 9 2
81
94cos
92cos
9cos that show Hence c)
sin sin 2sin cos2 2
A B A BA B
2
cos2
cos2coscos BABABA
cos cos 2sin sin2 2
A B A BA B
18765432 roots appear in conjugate pairs
2 4 6 82cos 2cos 2cos 2cos 19 9 9 9
(2 sine half sum cos half diff)
(2 cos half sum cos half diff)
(minus 2 sine half sum sine half diff)
3 7 12cos cos 2cos cos9 9 9 9 2
2 4 6 8 1cos cos cos cos9 9 9 9 2
3 7 12cos cos 2cos cos9 9 9 9 2
3 7 12cos cos 2cos cos9 9 9 9 2
3 7 1cos cos cos9 9 9 4
3 7 12cos cos 2cos cos9 9 9 9 2
3 7 1cos cos cos9 9 9 4
5 2 1cos 2cos cos9 9 9 4
3 7 12cos cos 2cos cos9 9 9 9 2
3 7 1cos cos cos9 9 9 4
5 2 1cos 2cos cos9 9 9 4
2 5 1cos cos cos9 9 9 8
3 7 12cos cos 2cos cos9 9 9 9 2
3 7 1cos cos cos9 9 9 4
5 2 1cos 2cos cos9 9 9 4
2 5 1cos cos cos9 9 9 8
5 4But cos cos9 9
3 7 12cos cos 2cos cos9 9 9 9 2
3 7 1cos cos cos9 9 9 4
5 2 1cos 2cos cos9 9 9 4
2 5 1cos cos cos9 9 9 8
5 4But cos cos9 9
2 4 1cos cos cos9 9 9 8
3 7 12cos cos 2cos cos9 9 9 9 2
3 7 1cos cos cos9 9 9 4
5 2 1cos 2cos cos9 9 9 4
2 5 1cos cos cos9 9 9 8
5 4But cos cos9 9
2 4 1cos cos cos9 9 9 8
2 4 1cos cos cos9 9 9 8
OR
9
8 2 7 3 6 4 5
1
1
z
z z z z z z z z z
OR
9
8 2 7 3 6 4 5
1
1
z
z z z z z z z z z
2 2
2 2
2 41 2cos 1 2cos 19 9
6 8 2cos 1 2cos 19 9
z z z z z
z z z z
OR
9
8 2 7 3 6 4 5
1
1
z
z z z z z z z z z
2 2
2 2
2 41 2cos 1 2cos 19 9
6 8 2cos 1 2cos 19 9
z z z z z
z z z z
2 2
2 2
2 41 2cos 1 2cos 19 9
8 1 2cos 19
z z z z z
z z z z
OR
9
8 2 7 3 6 4 5
1
1
z
z z z z z z z z z
2 2
2 2
2 41 2cos 1 2cos 19 9
6 8 2cos 1 2cos 19 9
z z z z z
z z z z
2 2
2 2
2 41 2cos 1 2cos 19 9
8 1 2cos 19
z z z z z
z z z z
Let z i
OR
9
8 2 7 3 6 4 5
1
1
z
z z z z z z z z z
2 2
2 2
2 41 2cos 1 2cos 19 9
6 8 2cos 1 2cos 19 9
z z z z z
z z z z
2 2
2 2
2 41 2cos 1 2cos 19 9
8 1 2cos 19
z z z z z
z z z z
Let z i
9 2 4 81 1 2cos 2cos 2cos9 9 9
i i i i i i
9 2 4 81 1 2cos 2cos 2cos9 9 9
i i i i i i
9 2 4 81 1 2cos 2cos 2cos9 9 9
i i i i i i
4 2 4 81 1 2cos 2cos 2cos9 9 9
i i i
9 2 4 81 1 2cos 2cos 2cos9 9 9
i i i i i i
4 2 4 81 1 2cos 2cos 2cos9 9 9
i i i
2 4 81 8cos cos cos9 9 9
9 2 4 81 1 2cos 2cos 2cos9 9 9
i i i i i i
4 2 4 81 1 2cos 2cos 2cos9 9 9
i i i
2 4 81 8cos cos cos9 9 9
2 41 8cos cos cos9 9 9
9 2 4 81 1 2cos 2cos 2cos9 9 9
i i i i i i
4 2 4 81 1 2cos 2cos 2cos9 9 9
i i i
2 4 81 8cos cos cos9 9 9
2 4 1cos cos cos9 9 9 8
2 41 8cos cos cos9 9 9
9 2 4 81 1 2cos 2cos 2cos9 9 9
i i i i i i
4 2 4 81 1 2cos 2cos 2cos9 9 9
i i i
2 4 81 8cos cos cos9 9 9
2 4 1cos cos cos9 9 9 8
Exercise 4J; 1 to 4, 7ac
2 41 8cos cos cos9 9 9