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X2 T02 02 complex factors

N
Nigel Simmons

The document discusses factorizing a polynomial P(x) = 4x^4 + 8x^3 + 5x^2 + x - 3 over the complex field. It is given that P(x) has two rational zeros. The zeros are found to be 1/2 and -3/2. P(x) is then factorized as (2x - 1)(2x + 3)(x^2 + x + 1) and further factorized into complex factors.

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Factorising Into Complex Factors e.g. 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.
Factorising Into Complex Factors e.g. 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  1  1  1 1 P   4   8   5    3   2   16   8   4  2 0
Factorising Into Complex Factors e.g. 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  1  1  1 1 P   4   8   5    3   2   16   8   4  2 0  2 x  1 is a factor
Factorising Into Complex Factors e.g. 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  1  1  1 1 P   4   8   5    3   2   16   8   4  2 0  2 x  1 is a factor P x   2 x  12 x 3  3
Factorising Into Complex Factors e.g. 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  1  1  1 1 P   4   8   5    3   2   16   8   4  2 0  2 x  1 is a factor P x   2 x  12 x 3  5 x  3
Factorising Into Complex Factors e.g. 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  1  1  1 1 P   4   8   5    3   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   2  27   5 9   5  3   3 P         2  8   4  2 0
  3   2  27   5 9   5  3   3 P         2  8   4  2 0  2 x  3 is a factor
  3   2  27   5 9   5  3   3 P         2  8   4  2 0  2 x  3 is a factor 1 3  rational zeros are and - 2 2
  3   2  27   5 9   5  3   3 P         2  8   4  2 0  2 x  3 is a factor 1 3  rational zeros are and - 2 2 P x   2 x  12 x  3x 2  1
  3   2  27   5 9   5  3   3 P         2  8   4  2 0  2 x  3 is a factor 1 3 1 3  2 x  6 x  rational zeros are and - 2 2 1 2 x  1  2 x P x   2 x  12 x  3x 2  1
  3   2  27   5 9   5  3   3 P         2  8   4  2 0  2 x  3 is a factor 1 3 1 3  2 x  6 x  rational zeros are and - 2 2 1 2 x  1  2 x P x   2 x  12 x  3x 2  1 4x  1 3  ? x  3 x
  3   2  27   5 9   5  3   3 P         2  8   4  2 0  2 x  3 is a factor 1 3 1 3  2 x  6 x  rational zeros are and - 2 2 1 2 x  1  2 x P x   2 x  12 x  3x 2  x  1 4x  1 3  ? x  3 x
  3   2  27   5 9   5  3   3 P         2  8   4  2 0  2 x  3 is a factor 1 3 1 3  2 x  6 x  rational zeros are and - 2 2 1 2 x  1  2 x P x   2 x  12 x  3x 2  x  1 4x  1  3 2  1 3  ? x  3 x  2 x  12 x  3 x      2  4
  3   2  27   5 9   5  3   3 P         2  8   4  2 0  2 x  3 is a factor 1 3 1 3  2 x  6 x  rational zeros are and - 2 2 1 2 x  1  2 x P x   2 x  12 x  3x 2  x  1 4x  1  3 2  1 3  ? x  3 x  2 x  12 x  3 x      2  4  1 3  1 3   2 x  12 x  3 x   i  x   i  2 2  2 2 
  3   2  27   5 9   5  3   3 P         2  8   4  2 0  2 x  3 is a factor 1 3 1 3  2 x  6 x  rational zeros are and - 2 2 1 2 x  1  2 x P x   2 x  12 x  3x 2  x  1 4x  1  3 2  1 3  ? x  3 x  2 x  12 x  3 x      2  4  1 3  1 3   2 x  12 x  3 x   i  x   i  2 2  2 2  Exercise 5C; 1 to 15 odds, 16 to 20 all

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X2 T02 02 complex factors

  • 1. Factorising Into Complex Factors e.g. 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.
  • 2. Factorising Into Complex Factors e.g. 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  1  1  1 1 P   4   8   5    3   2   16   8   4  2 0
  • 3. Factorising Into Complex Factors e.g. 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  1  1  1 1 P   4   8   5    3   2   16   8   4  2 0  2 x  1 is a factor
  • 4. Factorising Into Complex Factors e.g. 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  1  1  1 1 P   4   8   5    3   2   16   8   4  2 0  2 x  1 is a factor P x   2 x  12 x 3  3
  • 5. Factorising Into Complex Factors e.g. 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  1  1  1 1 P   4   8   5    3   2   16   8   4  2 0  2 x  1 is a factor P x   2 x  12 x 3  5 x  3
  • 6. Factorising Into Complex Factors e.g. 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  1  1  1 1 P   4   8   5    3   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
  • 7.   3   2  27   5 9   5  3   3 P         2  8   4  2 0
  • 8.   3   2  27   5 9   5  3   3 P         2  8   4  2 0  2 x  3 is a factor
  • 9.   3   2  27   5 9   5  3   3 P         2  8   4  2 0  2 x  3 is a factor 1 3  rational zeros are and - 2 2
  • 10.   3   2  27   5 9   5  3   3 P         2  8   4  2 0  2 x  3 is a factor 1 3  rational zeros are and - 2 2 P x   2 x  12 x  3x 2  1
  • 11.   3   2  27   5 9   5  3   3 P         2  8   4  2 0  2 x  3 is a factor 1 3 1 3  2 x  6 x  rational zeros are and - 2 2 1 2 x  1  2 x P x   2 x  12 x  3x 2  1
  • 12.   3   2  27   5 9   5  3   3 P         2  8   4  2 0  2 x  3 is a factor 1 3 1 3  2 x  6 x  rational zeros are and - 2 2 1 2 x  1  2 x P x   2 x  12 x  3x 2  1 4x  1 3  ? x  3 x
  • 13.   3   2  27   5 9   5  3   3 P         2  8   4  2 0  2 x  3 is a factor 1 3 1 3  2 x  6 x  rational zeros are and - 2 2 1 2 x  1  2 x P x   2 x  12 x  3x 2  x  1 4x  1 3  ? x  3 x
  • 14.   3   2  27   5 9   5  3   3 P         2  8   4  2 0  2 x  3 is a factor 1 3 1 3  2 x  6 x  rational zeros are and - 2 2 1 2 x  1  2 x P x   2 x  12 x  3x 2  x  1 4x  1  3 2  1 3  ? x  3 x  2 x  12 x  3 x      2  4
  • 15.   3   2  27   5 9   5  3   3 P         2  8   4  2 0  2 x  3 is a factor 1 3 1 3  2 x  6 x  rational zeros are and - 2 2 1 2 x  1  2 x P x   2 x  12 x  3x 2  x  1 4x  1  3 2  1 3  ? x  3 x  2 x  12 x  3 x      2  4  1 3  1 3   2 x  12 x  3 x   i  x   i  2 2  2 2 
  • 16.   3   2  27   5 9   5  3   3 P         2  8   4  2 0  2 x  3 is a factor 1 3 1 3  2 x  6 x  rational zeros are and - 2 2 1 2 x  1  2 x P x   2 x  12 x  3x 2  x  1 4x  1  3 2  1 3  ? x  3 x  2 x  12 x  3 x      2  4  1 3  1 3   2 x  12 x  3 x   i  x   i  2 2  2 2  Exercise 5C; 1 to 15 odds, 16 to 20 all