X2 T01 08 factorising complex expressions (2010)

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  • Factorising Complex Expressions

  • Factorising Complex Expressions

    If a polynomials coefficients are all real then the roots will appear in complex conjugate pairs.

  • Factorising Complex Expressions

    If a polynomials 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 polynomials 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 polynomials 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 polynomials 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 polynomials 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 polynomials 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 polynomials 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

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    numbersRealover factorised 0433 2 zzz

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    numbersRealover factorised 0433 2 zzz 02233 izizzz

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    0433 2 zzz 02233 izizzz

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    numbersRealover factorised numbersComplex over factorised

    izz 2or 3

    0433 2 zzz 02233 izizzz

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    izz 2or 3 5832 Factorise 23 xxxiii

    0433 2 zzz 02233 izizzz

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    numbersRealover factorised numbersComplex over factorised

    izz 2or 3 5832 Factorise 23 xxxiii

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    as it is a cubic it must have a real factor

  • 012 24 zzii 043 22 zz

    numbersRealover factorised numbersComplex over factorised

    izz 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 factorised

    izz 2or 3 5832 Factorise 23 xxxiii

    52125832 223 xxxxxx 054

    43

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    as it is a cubic it must have a real factor

  • 012 24 zzii 043 22 zz

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    izz 2or 3 5832 Factorise 23 xxxiii

    52125832 223 xxxxxx 054

    43

    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 factorised

    izz 2or 3 5832 Factorise 23 xxxiii

    52125832 223 xxxxxx 054

    43

    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

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    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

  • 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

    zzzzzz

  • 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

    zzzzzz

  • 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

    zzzzzz

    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

    zzzzzz

    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

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

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    k

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    019 z01 8765432 rootsofsum

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

    2sin9

    2cosLet

    HSC9961

    9 kz

    i

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

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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 pairs2 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 pairs2 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 pairs2 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 22 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 22 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 22 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 22 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 2cos

    9 9 9i 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 2cos

    9 9 9i 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 2cos

    9 9 9i 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 2cos

    9 9 9i 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 2cos

    9 9 9i 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 2cos

    9 9 9i 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

    Slide Number 1Slide Number 2Slide Number 3Slide Number 4Slide Number 5Slide Number 6Slide Number 7Slide Number 8Slide Number 9Slide Number 10Slide Number 11Slide Number 12Slide Number 13Slide Number 14Slide Number 15Slide Number 16Slide Number 17Slide Number 18Slide Number 19Slide Number 20Slide Number 21Slide Number 22Slide Number 23Slide Number 24Slide Number 25Slide Number 26Slide Number 27Slide Number 28Slide Number 29Slide Number 30Slide Number 31Slide Number 32Slide Number 33Slide Number 34Slide Number 35Slide Number 36Slide Number 37Slide Number 38Slide Number 39Slide Number 40Slide Number 41Slide Number 42Slide Number 43Slide Number 44Slide Number 45Slide Number 46Slide Number 47Slide Number 48Slide Number 49Slide Number 50Slide Number 51Slide Number 52Slide Number 53Slide Number 54Slide Number 55Slide Number 56Slide Number 57Slide Number 58Slide Number 59Slide Number 60Slide Number 61Slide Number 62Slide Number 63Slide Number 64Slide Number 65Slide Number 66Slide Number 67Slide Number 68Slide Number 69Slide Number 70