X2 t01 08 factorising complex expressions (2012)
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Factorising Complex Expressions
![Page 2: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/2.jpg)
Factorising Complex Expressions
If a polynomial’s coefficients are all real then the roots will appear in complex conjugate pairs.
![Page 3: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/3.jpg)
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;
![Page 4: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/4.jpg)
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
![Page 5: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/5.jpg)
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
![Page 6: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/6.jpg)
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
![Page 7: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/7.jpg)
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
![Page 8: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/8.jpg)
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
![Page 9: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/9.jpg)
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
![Page 10: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/10.jpg)
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![Page 12: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/12.jpg)
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![Page 14: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/14.jpg)
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![Page 15: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/15.jpg)
012 24 zzii 043 22 zz
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![Page 16: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/16.jpg)
012 24 zzii 043 22 zz
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![Page 17: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/17.jpg)
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![Page 18: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/18.jpg)
012 24 zzii 043 22 zz
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as it is a cubic it must have a real factor
![Page 19: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/19.jpg)
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as it is a cubic it must have a real factor
![Page 20: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/20.jpg)
012 24 zzii 043 22 zz
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as it is a cubic it must have a real factor
![Page 21: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/21.jpg)
012 24 zzii 043 22 zz
numbersRealover factorised
numbersComplex over factorisedizz 2or 3
5832 Factorise 23 xxxiii
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02233 izizzz
as it is a cubic it must have a real factor
![Page 22: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/22.jpg)
012 24 zzii 043 22 zz
numbersRealover factorised
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as it is a cubic it must have a real factor
![Page 23: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/23.jpg)
0 1 2345 zzzzziv
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0 1 2345 zzzzziv
111 Now 23456 zzzzzzz
![Page 25: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/25.jpg)
0 1 2345 zzzzziv
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2 0, 1, 2,36
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![Page 26: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/26.jpg)
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![Page 27: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/27.jpg)
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![Page 28: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/28.jpg)
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![Page 29: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/29.jpg)
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![Page 30: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/30.jpg)
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![Page 31: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/31.jpg)
integeran is where,01 ofsolution a is that Show a)9
2sin9
2cosLet
HSC9961
9 kz
i
v
k
![Page 32: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/32.jpg)
integeran is where,01 ofsolution a is that Show a)9
2sin9
2cosLet
HSC9961
9 kz
i
v
k
19 z
![Page 33: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/33.jpg)
integeran is where,01 ofsolution a is that Show a)9
2sin9
2cosLet
HSC9961
9 kz
i
v
k
19 z8,7,6,5,4,3,2,1,0
92
kkcisz
![Page 34: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/34.jpg)
integeran is where,01 ofsolution a is that Show a)9
2sin9
2cosLet
HSC9961
9 kz
i
v
k
19 z8,7,6,5,4,3,2,1,0
92
kkcisz
k
cisz
92
![Page 35: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/35.jpg)
integeran is where,01 ofsolution a is that Show a)9
2sin9
2cosLet
HSC9961
9 kz
i
v
k
19 z8,7,6,5,4,3,2,1,0
92
kkcisz
k
cisz
92
kz
![Page 36: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/36.jpg)
integeran is where,01 ofsolution a is that Show a)9
2sin9
2cosLet
HSC9961
9 kz
i
v
k
19 z8,7,6,5,4,3,2,1,0
92
kkcisz
k
cisz
92
kz 1 that Prove b) 8765432
![Page 37: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/37.jpg)
integeran is where,01 ofsolution a is that Show a)9
2sin9
2cosLet
HSC9961
9 kz
i
v
k
19 z8,7,6,5,4,3,2,1,0
92
kkcisz
k
cisz
92
kz 1 that Prove b) 8765432
019 z
![Page 38: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/38.jpg)
integeran is where,01 ofsolution a is that Show a)9
2sin9
2cosLet
HSC9961
9 kz
i
v
k
19 z8,7,6,5,4,3,2,1,0
92
kkcisz
k
cisz
92
kz 1 that Prove b) 8765432
019 z01 8765432
![Page 39: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/39.jpg)
integeran is where,01 ofsolution a is that Show a)9
2sin9
2cosLet
HSC9961
9 kz
i
v
k
19 z8,7,6,5,4,3,2,1,0
92
kkcisz
k
cisz
92
kz 1 that Prove b) 8765432
019 z01 8765432 rootsofsum
![Page 40: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/40.jpg)
integeran is where,01 ofsolution a is that Show a)9
2sin9
2cosLet
HSC9961
9 kz
i
v
k
19 z8,7,6,5,4,3,2,1,0
92
kkcisz
k
cisz
92
kz 1 that Prove b) 8765432
019 z01 8765432 rootsofsum
18765432
![Page 41: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/41.jpg)
81
94cos
92cos
9cos that show Hence c)
![Page 42: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/42.jpg)
81
94cos
92cos
9cos that show Hence c)
sin sin 2sin cos2 2
A B A BA B
![Page 43: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/43.jpg)
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)
![Page 44: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/44.jpg)
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)
![Page 45: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/45.jpg)
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)
![Page 46: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/46.jpg)
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)
![Page 47: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/47.jpg)
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)
![Page 48: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/48.jpg)
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
(2 sine half sum cos half diff)
(2 cos half sum cos half diff)
(minus 2 sine half sum sine half diff)
![Page 49: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/49.jpg)
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)
![Page 50: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/50.jpg)
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)
![Page 51: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/51.jpg)
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)
2 4 6 8 1cos cos cos cos9 9 9 9 2
![Page 52: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/52.jpg)
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
![Page 53: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/53.jpg)
3 7 12cos cos 2cos cos9 9 9 9 2
![Page 54: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/54.jpg)
3 7 12cos cos 2cos cos9 9 9 9 2
3 7 1cos cos cos9 9 9 4
![Page 55: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/55.jpg)
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
![Page 56: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/56.jpg)
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
![Page 57: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/57.jpg)
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
![Page 58: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/58.jpg)
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
![Page 59: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/59.jpg)
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
![Page 60: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/60.jpg)
OR
9
8 2 7 3 6 4 5
1
1
z
z z z z z z z z z
![Page 61: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/61.jpg)
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
![Page 62: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/62.jpg)
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
![Page 63: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/63.jpg)
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
![Page 64: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/64.jpg)
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
![Page 65: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/65.jpg)
9 2 4 81 1 2cos 2cos 2cos9 9 9
i i i i i i
![Page 66: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/66.jpg)
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
![Page 67: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/67.jpg)
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
![Page 68: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/68.jpg)
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
![Page 69: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/69.jpg)
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
![Page 70: X2 t01 08 factorising complex expressions (2012)](https://reader036.fdocuments.in/reader036/viewer/2022081401/5596a5c51a28ab335b8b477a/html5/thumbnails/70.jpg)
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