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X2 t02 01 factorising complex expressions (2013)

N
Nigel Simmons

The document discusses factorizing complex expressions. The main points are: - If a polynomial's coefficients are real, its roots will appear in complex conjugate pairs. - Any polynomial of degree n can be factorized into a mixture of quadratic and linear factors over real numbers, or into n linear factors over complex numbers. - Odd degree polynomials must have at least one real root. - Examples of factorizing polynomials over both real and complex numbers are provided.

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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; • 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 field NOTE: odd ordered polynomials must have a real root
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 NOTE: odd ordered polynomials must have a real root e.g . i  x 2  2 x  2
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 NOTE: odd ordered polynomials must have a real root e.g . i  x 2  2 x  2   x  12  1
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 NOTE: odd ordered polynomials must have a real root e.g . i  x 2  2 x  2   x  12  1   x  1  i  x  1  i 
ii  z 4  z 2  12  0
ii  z 4  z 2  12  0 z 2  3z 2  4   0
ii  z 4  z 2  12  0 z  3z  4  0 z  3 z  3z  4  0 2 2 2 factorised over Real numbers
ii  z 4  z 2  12  0 z  3z  4  0 factorised over Real numbers z  3 z  3z  4  0 z  3 z  3 z  2i  z  2i   0 factorised over Complex numbers 2 2 2
ii  z 4  z 2  12  0 z  3z  4  0 factorised over Real numbers z  3 z  3z  4  0 z  3 z  3 z  2i  z  2i   0 factorised over Complex numbers 2 2 2 z   3 or z  2i
ii  z 4  z 2  12  0 z  3z  4  0 factorised over Real numbers z  3 z  3z  4  0 z  3 z  3 z  2i  z  2i   0 factorised over Complex numbers 2 2 2 z   3 or z  2i If (x – a) is a factor of P(x), then P(a) = 0 b If (ax – b) is a factor of P(x), then P  = 0 a
ii  z 4  z 2  12  0 z  3z  4  0 factorised over Real numbers z  3 z  3z  4  0 z  3 z  3 z  2i  z  2i   0 factorised over Complex numbers 2 2 2 z   3 or z  2i If (x – a) is a factor of P(x), then P(a) = 0 b If (ax – b) is a factor of P(x), then P  = 0 a iii  Factorise 2 x 3  3x 2  8 x  5
ii  z 4  z 2  12  0 z  3z  4  0 factorised over Real numbers z  3 z  3z  4  0 z  3 z  3 z  2i  z  2i   0 factorised over Complex numbers 2 2 2 z   3 or z  2i If (x – a) is a factor of P(x), then P(a) = 0 b If (ax – b) is a factor of P(x), then P  = 0 a iii  Factorise 2 x 3  3x 2  8 x  5 as it is a cubic it must have a real factor
ii  z 4  z 2  12  0 z  3z  4  0 factorised over Real numbers z  3 z  3z  4  0 z  3 z  3 z  2i  z  2i   0 factorised over Complex numbers 2 2 2 z   3 or z  2i If (x – a) is a factor of P(x), then P(a) = 0 b If (ax – b) is a factor of P(x), then P  = 0 a iii  Factorise 2 x 3  3x 2  8 x  5 as it is a cubic it must have a real factor 3 2 1  1 1    1 5 P   2    3   8   2  2  2  2 1 3    45 4 4 0
ii  z 4  z 2  12  0 z  3z  4  0 factorised over Real numbers z  3 z  3z  4  0 z  3 z  3 z  2i  z  2i   0 factorised over Complex numbers 2 2 2 z   3 or z  2i If (x – a) is a factor of P(x), then P(a) = 0 b If (ax – b) is a factor of P(x), then P  = 0 a iii  Factorise 2 x 3  3x 2  8 x  5 as it is a cubic it must have a real factor 3 2 1  1 1    1 5  2 x3  3x 2  8 x  5 P   2    3   8   2  2  2  2   2 x  1  x 2  2 x  5  1 3    45 4 4 0
ii  z 4  z 2  12  0 z  3z  4  0 factorised over Real numbers z  3 z  3z  4  0 z  3 z  3 z  2i  z  2i   0 factorised over Complex numbers 2 2 2 z   3 or z  2i If (x – a) is a factor of P(x), then P(a) = 0 b If (ax – b) is a factor of P(x), then P  = 0 a iii  Factorise 2 x 3  3x 2  8 x  5 as it is a cubic it must have a real factor 3 2 1  1 1    1 5  2 x3  3x 2  8 x  5 P   2    3   8   2  2  2  2   2 x  1  x 2  2 x  5  1 3    45 2  2 x  1  x  1  4 4 4 0  
ii  z 4  z 2  12  0 z  3z  4  0 factorised over Real numbers z  3 z  3z  4  0 z  3 z  3 z  2i  z  2i   0 factorised over Complex numbers 2 2 2 z   3 or z  2i If (x – a) is a factor of P(x), then P(a) = 0 b If (ax – b) is a factor of P(x), then P  = 0 a iii  Factorise 2 x 3  3x 2  8 x  5 as it is a cubic it must have a real factor 3 2 1  1 1    1 5  2 x3  3x 2  8 x  5 P   2    3   8   2  2  2  2   2 x  1  x 2  2 x  5  1 3    45 2  2 x  1  x  1  4 4 4  2 x  1 x  1  2i  x  1  2i  0  
(iv) Given that P x   4 x 4  8 x 3  5 x 2  x  3 has two rational zeros, find these zeros and factorise P(x) over the complex field.
(iv) Given that P x   4 x 4  8 x 3  5 x 2  x  3 has two rational zeros, find these zeros and factorise P(x) over the complex field.  1   4 1   8 1   5 1   1  3 P         2   16   8   4  2 0  2 x  1 is a factor
(iv) Given that P x   4 x 4  8 x 3  5 x 2  x  3 has two rational zeros, find these zeros and factorise P(x) over the complex field.  1   4 1   8 1   5 1   1  3 P         2   16   8   4  2 0  2 x  1 is a factor P x   2 x  12 x 3  3
(iv) Given that P x   4 x 4  8 x 3  5 x 2  x  3 has two rational zeros, find these zeros and factorise P(x) over the complex field.  1   4 1   8 1   5 1   1  3 P         2   16   8   4  2 0  2 x  1 is a factor P x   2 x  12 x 3  5 x  3
(iv) Given that P x   4 x 4  8 x 3  5 x 2  x  3 has two rational zeros, find these zeros and factorise P(x) over the complex field.  1   4 1   8 1   5 1   1  3 P         2   16   8   4  2 0  2 x  1 is a factor P x   2 x  12 x 3 5x 2  5 x  3
(iv) Given that P x   4 x 4  8 x 3  5 x 2  x  3 has two rational zeros, find these zeros and factorise P(x) over the complex field.  1   4 1   8 1   5 1   1  3 P         2   16   8   4  2 0  2 x  1 is a factor P x   2 x  12 x 3 5x 2  5 x  3  3    27   9   3  P    2   5   5    3  2  8   4  2 0
(iv) Given that P x   4 x 4  8 x 3  5 x 2  x  3 has two rational zeros, find these zeros and factorise P(x) over the complex field.  1   4 1   8 1   5 1   1  3 P         2   16   8   4  2 0  2 x  1 is a factor P x   2 x  12 x 3 5x 2  5 x  3  3    27   9   3  P    2   5   5    3  2  8   4  2 0  2 x  3 is a factor 3 1  rational zeros are and 2 2
P x   4 x 4  8 x 3  5 x 2  x  3
P x   4 x 4  8 x 3  5 x 2  x  3   2 x  1 2 x  3  x 2  1
P x   4 x 4  8 x 3  5 x 2  x  3   2 x  1 2 x  3  x 2  1 1 3  2 x  6 x 1 2 x  1  2 x
P x   4 x 4  8 x 3  5 x 2  x  3   2 x  1 2 x  3  x 2  1 1 3  2 x  6 x 1 2 x  1  2 x 4x  1 3  ? x  3 x
P x   4 x 4  8 x 3  5 x 2  x  3   2 x  1 2 x  3  x 2 x  1 1 3  2 x  6 x 1 2 x  1  2 x 4x  1 3  ? x  3 x
P x   4 x 4  8 x 3  5 x 2  x  3   2 x  1 2 x  3  x 2 x  1 2  1  3  2 x  12 x  3 x     2  4  1 3  2 x  6 x 1 2 x  1  2 x 4x  1 3  ? x  3 x
P x   4 x 4  8 x 3  5 x 2  x  3 1 3  2 x  6 x   2 x  1 2 x  3  x 2 x  1 1 2 x  1  2 x 2  1  3 4x  2 x  12 x  3 x     2  4  1 3  ? x  3 x  1 3  1 3    2 x  12 x  3 x   i  x   i 2 2  2 2  
P x   4 x 4  8 x 3  5 x 2  x  3 1 3  2 x  6 x   2 x  1 2 x  3  x 2 x  1 1 2 x  1  2 x 2  1  3 4x  2 x  12 x  3 x     2  4  1 3  ? x  3 x  1 3  1 3    2 x  12 x  3 x   i  x   i 2 2  2 2   Cambridge: Exercise 5A; 1b, 2, 3, 5, 6, 7b, 8, 9, 10, 12 to 15

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X2 t02 01 factorising complex expressions (2013)

  • 2. Factorising Complex Expressions If a polynomial’s coefficients are all real then the roots will appear in complex conjugate pairs.
  • 3. 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
  • 4. 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
  • 5. 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 NOTE: odd ordered polynomials must have a real root
  • 6. 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 NOTE: odd ordered polynomials must have a real root e.g . i  x 2  2 x  2
  • 7. 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 NOTE: odd ordered polynomials must have a real root e.g . i  x 2  2 x  2   x  12  1
  • 8. 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 NOTE: odd ordered polynomials must have a real root e.g . i  x 2  2 x  2   x  12  1   x  1  i  x  1  i 
  • 9. ii  z 4  z 2  12  0
  • 10. ii  z 4  z 2  12  0 z 2  3z 2  4   0
  • 11. ii  z 4  z 2  12  0 z  3z  4  0 z  3 z  3z  4  0 2 2 2 factorised over Real numbers
  • 12. ii  z 4  z 2  12  0 z  3z  4  0 factorised over Real numbers z  3 z  3z  4  0 z  3 z  3 z  2i  z  2i   0 factorised over Complex numbers 2 2 2
  • 13. ii  z 4  z 2  12  0 z  3z  4  0 factorised over Real numbers z  3 z  3z  4  0 z  3 z  3 z  2i  z  2i   0 factorised over Complex numbers 2 2 2 z   3 or z  2i
  • 14. ii  z 4  z 2  12  0 z  3z  4  0 factorised over Real numbers z  3 z  3z  4  0 z  3 z  3 z  2i  z  2i   0 factorised over Complex numbers 2 2 2 z   3 or z  2i If (x – a) is a factor of P(x), then P(a) = 0 b If (ax – b) is a factor of P(x), then P  = 0 a
  • 15. ii  z 4  z 2  12  0 z  3z  4  0 factorised over Real numbers z  3 z  3z  4  0 z  3 z  3 z  2i  z  2i   0 factorised over Complex numbers 2 2 2 z   3 or z  2i If (x – a) is a factor of P(x), then P(a) = 0 b If (ax – b) is a factor of P(x), then P  = 0 a iii  Factorise 2 x 3  3x 2  8 x  5
  • 16. ii  z 4  z 2  12  0 z  3z  4  0 factorised over Real numbers z  3 z  3z  4  0 z  3 z  3 z  2i  z  2i   0 factorised over Complex numbers 2 2 2 z   3 or z  2i If (x – a) is a factor of P(x), then P(a) = 0 b If (ax – b) is a factor of P(x), then P  = 0 a iii  Factorise 2 x 3  3x 2  8 x  5 as it is a cubic it must have a real factor
  • 17. ii  z 4  z 2  12  0 z  3z  4  0 factorised over Real numbers z  3 z  3z  4  0 z  3 z  3 z  2i  z  2i   0 factorised over Complex numbers 2 2 2 z   3 or z  2i If (x – a) is a factor of P(x), then P(a) = 0 b If (ax – b) is a factor of P(x), then P  = 0 a iii  Factorise 2 x 3  3x 2  8 x  5 as it is a cubic it must have a real factor 3 2 1  1 1    1 5 P   2    3   8   2  2  2  2 1 3    45 4 4 0
  • 18. ii  z 4  z 2  12  0 z  3z  4  0 factorised over Real numbers z  3 z  3z  4  0 z  3 z  3 z  2i  z  2i   0 factorised over Complex numbers 2 2 2 z   3 or z  2i If (x – a) is a factor of P(x), then P(a) = 0 b If (ax – b) is a factor of P(x), then P  = 0 a iii  Factorise 2 x 3  3x 2  8 x  5 as it is a cubic it must have a real factor 3 2 1  1 1    1 5  2 x3  3x 2  8 x  5 P   2    3   8   2  2  2  2   2 x  1  x 2  2 x  5  1 3    45 4 4 0
  • 19. ii  z 4  z 2  12  0 z  3z  4  0 factorised over Real numbers z  3 z  3z  4  0 z  3 z  3 z  2i  z  2i   0 factorised over Complex numbers 2 2 2 z   3 or z  2i If (x – a) is a factor of P(x), then P(a) = 0 b If (ax – b) is a factor of P(x), then P  = 0 a iii  Factorise 2 x 3  3x 2  8 x  5 as it is a cubic it must have a real factor 3 2 1  1 1    1 5  2 x3  3x 2  8 x  5 P   2    3   8   2  2  2  2   2 x  1  x 2  2 x  5  1 3    45 2  2 x  1  x  1  4 4 4 0  
  • 20. ii  z 4  z 2  12  0 z  3z  4  0 factorised over Real numbers z  3 z  3z  4  0 z  3 z  3 z  2i  z  2i   0 factorised over Complex numbers 2 2 2 z   3 or z  2i If (x – a) is a factor of P(x), then P(a) = 0 b If (ax – b) is a factor of P(x), then P  = 0 a iii  Factorise 2 x 3  3x 2  8 x  5 as it is a cubic it must have a real factor 3 2 1  1 1    1 5  2 x3  3x 2  8 x  5 P   2    3   8   2  2  2  2   2 x  1  x 2  2 x  5  1 3    45 2  2 x  1  x  1  4 4 4  2 x  1 x  1  2i  x  1  2i  0  
  • 21. (iv) Given that P x   4 x 4  8 x 3  5 x 2  x  3 has two rational zeros, find these zeros and factorise P(x) over the complex field.
  • 22. (iv) Given that P x   4 x 4  8 x 3  5 x 2  x  3 has two rational zeros, find these zeros and factorise P(x) over the complex field.  1   4 1   8 1   5 1   1  3 P         2   16   8   4  2 0  2 x  1 is a factor
  • 23. (iv) Given that P x   4 x 4  8 x 3  5 x 2  x  3 has two rational zeros, find these zeros and factorise P(x) over the complex field.  1   4 1   8 1   5 1   1  3 P         2   16   8   4  2 0  2 x  1 is a factor P x   2 x  12 x 3  3
  • 24. (iv) Given that P x   4 x 4  8 x 3  5 x 2  x  3 has two rational zeros, find these zeros and factorise P(x) over the complex field.  1   4 1   8 1   5 1   1  3 P         2   16   8   4  2 0  2 x  1 is a factor P x   2 x  12 x 3  5 x  3
  • 25. (iv) Given that P x   4 x 4  8 x 3  5 x 2  x  3 has two rational zeros, find these zeros and factorise P(x) over the complex field.  1   4 1   8 1   5 1   1  3 P         2   16   8   4  2 0  2 x  1 is a factor P x   2 x  12 x 3 5x 2  5 x  3
  • 26. (iv) Given that P x   4 x 4  8 x 3  5 x 2  x  3 has two rational zeros, find these zeros and factorise P(x) over the complex field.  1   4 1   8 1   5 1   1  3 P         2   16   8   4  2 0  2 x  1 is a factor P x   2 x  12 x 3 5x 2  5 x  3  3    27   9   3  P    2   5   5    3  2  8   4  2 0
  • 27. (iv) Given that P x   4 x 4  8 x 3  5 x 2  x  3 has two rational zeros, find these zeros and factorise P(x) over the complex field.  1   4 1   8 1   5 1   1  3 P         2   16   8   4  2 0  2 x  1 is a factor P x   2 x  12 x 3 5x 2  5 x  3  3    27   9   3  P    2   5   5    3  2  8   4  2 0  2 x  3 is a factor 3 1  rational zeros are and 2 2
  • 28. P x   4 x 4  8 x 3  5 x 2  x  3
  • 29. P x   4 x 4  8 x 3  5 x 2  x  3   2 x  1 2 x  3  x 2  1
  • 30. P x   4 x 4  8 x 3  5 x 2  x  3   2 x  1 2 x  3  x 2  1 1 3  2 x  6 x 1 2 x  1  2 x
  • 31. P x   4 x 4  8 x 3  5 x 2  x  3   2 x  1 2 x  3  x 2  1 1 3  2 x  6 x 1 2 x  1  2 x 4x  1 3  ? x  3 x
  • 32. P x   4 x 4  8 x 3  5 x 2  x  3   2 x  1 2 x  3  x 2 x  1 1 3  2 x  6 x 1 2 x  1  2 x 4x  1 3  ? x  3 x
  • 33. P x   4 x 4  8 x 3  5 x 2  x  3   2 x  1 2 x  3  x 2 x  1 2  1  3  2 x  12 x  3 x     2  4  1 3  2 x  6 x 1 2 x  1  2 x 4x  1 3  ? x  3 x
  • 34. P x   4 x 4  8 x 3  5 x 2  x  3 1 3  2 x  6 x   2 x  1 2 x  3  x 2 x  1 1 2 x  1  2 x 2  1  3 4x  2 x  12 x  3 x     2  4  1 3  ? x  3 x  1 3  1 3    2 x  12 x  3 x   i  x   i 2 2  2 2  
  • 35. P x   4 x 4  8 x 3  5 x 2  x  3 1 3  2 x  6 x   2 x  1 2 x  3  x 2 x  1 1 2 x  1  2 x 2  1  3 4x  2 x  12 x  3 x     2  4  1 3  ? x  3 x  1 3  1 3    2 x  12 x  3 x   i  x   i 2 2  2 2   Cambridge: Exercise 5A; 1b, 2, 3, 5, 6, 7b, 8, 9, 10, 12 to 15