Upload
nigel-simmons
View
822
Download
1
Embed Size (px)
Citation preview
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
012 24 zzii
012 24 zzii 043 22 zz
012 24 zzii 043 22 zz
0433 2 zzz
012 24 zzii 043 22 zz
numbersRealover factorised 0433 2 zzz
012 24 zzii 043 22 zz
numbersRealover factorised 0433 2 zzz
02233 izizzz
012 24 zzii 043 22 zz
numbersRealover factorised
numbersComplex over factorised
0433 2 zzz
02233 izizzz
012 24 zzii 043 22 zz
numbersRealover factorised
numbersComplex over factorisedizz 2or 3
0433 2 zzz
02233 izizzz
012 24 zzii 043 22 zz
numbersRealover factorised
numbersComplex over factorisedizz 2or 3
5832 Factorise 23 xxxiii
0433 2 zzz
02233 izizzz
012 24 zzii 043 22 zz
numbersRealover factorised
numbersComplex over factorisedizz 2or 3
5832 Factorise 23 xxxiii
0433 2 zzz
02233 izizzz
as it is a cubic it must have a real factor
012 24 zzii 043 22 zz
numbersRealover factorised
numbersComplex over factorisedizz 2or 3
5832 Factorise 23 xxxiii
0
5443
41
5218
213
212
21 23
P
0433 2 zzz
02233 izizzz
as it is a cubic it must have a real factor
012 24 zzii 043 22 zz
numbersRealover factorised
numbersComplex over factorisedizz 2or 3
5832 Factorise 23 xxxiii
52125832 223 xxxxxx0
5443
41
5218
213
212
21 23
P
0433 2 zzz
02233 izizzz
as it is a cubic it must have a real factor
012 24 zzii 043 22 zz
numbersRealover factorised
numbersComplex over factorisedizz 2or 3
5832 Factorise 23 xxxiii
52125832 223 xxxxxx0
5443
41
5218
213
212
21 23
P
4112 2 xx
0433 2 zzz
02233 izizzz
as it is a cubic it must have a real factor
012 24 zzii 043 22 zz
numbersRealover factorised
numbersComplex over factorisedizz 2or 3
5832 Factorise 23 xxxiii
52125832 223 xxxxxx0
5443
41
5218
213
212
21 23
P
4112 2 xx ixixx 212112
0433 2 zzz
02233 izizzz
as it is a cubic it must have a real factor
0 1 2345 zzzzziv
0 1 2345 zzzzziv
111 Now 23456 zzzzzzz
0 1 2345 zzzzziv
111 Now 23456 zzzzzzz6And 1 0 has solutions;
2 0, 1, 2,36
zkz cis k
0 1 2345 zzzzziv
111 Now 23456 zzzzzzz
012345 zzzz
6And 1 0 has solutions;2 0, 1, 2,3
6
zkz cis k
0 1 2345 zzzzziv
111 Now 23456 zzzzzzz
012345 zzzz
6And 1 0 has solutions;2 0, 1, 2,3
6
zkz cis k
01
11 2345
z
zzzzz
0 1 2345 zzzzziv
111 Now 23456 zzzzzzz
012345 zzzz
1,016 zz
6And 1 0 has solutions;2 0, 1, 2,3
6
zkz cis k
01
11 2345
z
zzzzz
0 1 2345 zzzzziv
111 Now 23456 zzzzzzz
012345 zzzz
1,016 zz
6And 1 0 has solutions;2 0, 1, 2,3
6
zkz cis k
01
11 2345
z
zzzzz
2 2, , , ,3 3 3 3
z cis cis cis cis cis
0 1 2345 zzzzziv
111 Now 23456 zzzzzzz
012345 zzzz
1,016 zz
6And 1 0 has solutions;2 0, 1, 2,3
6
zkz cis k
01
11 2345
z
zzzzz
2 2, , , ,3 3 3 3
z cis cis cis cis cis
1 3 1 3 1 3 1 3, , , , 12 2 2 2 2 2 2 2
z i i i i
integeran is where,01 ofsolution a is that Show a)9
2sin9
2cosLet
HSC9961
9 kz
i
v
k
integeran is where,01 ofsolution a is that Show a)9
2sin9
2cosLet
HSC9961
9 kz
i
v
k
19 z
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
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
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
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
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
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
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
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
81
94cos
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
(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 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
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
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