X2 T01 08 factorising complex expressions (2011)

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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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

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02233 izizzz

012 24 zzii 043 22 zz

numbersRealover factorised

numbersComplex over factorisedizz 2or 3

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02233 izizzz

012 24 zzii 043 22 zz

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numbersComplex over factorisedizz 2or 3

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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

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5443

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5218

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0433 2 zzz

02233 izizzz

as it is a cubic it must have a real factor

012 24 zzii 043 22 zz

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5443

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5218

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0433 2 zzz

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as it is a cubic it must have a real factor

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5218

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212

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5218

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212

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4112 2 xx ixixx 212112

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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

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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

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9 kz

i

v

k

integeran is where,01 ofsolution a is that Show a)9

2sin9

2cosLet

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9 kz

i

v

k

19 z

integeran is where,01 ofsolution a is that Show a)9

2sin9

2cosLet

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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

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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

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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

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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

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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

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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

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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

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9 kz

i

v

k

19 z8,7,6,5,4,3,2,1,0

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kkcisz

k

cisz

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kz 1 that Prove b) 8765432

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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