11 x1 t15 06 roots & coefficients (2013)

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11 x1 t15 06 roots & coefficients (2013)

  1. 1. Roots and Coefficients
  2. 2. Roots and Coefficients Quadratics ax 2  bx  c  0
  3. 3. Roots and Coefficients Quadratics ax 2  bx  c  0 b    a
  4. 4. Roots and Coefficients Quadratics ax 2  bx  c  0 b    a c   a
  5. 5. Roots and Coefficients Quadratics ax 2  bx  c  0 b    a Cubics ax 3  bx 2  cx  d  0 c   a
  6. 6. Roots and Coefficients Quadratics ax 2  bx  c  0 b    a Cubics ax 3  bx 2  cx  d  0       b a c   a
  7. 7. Roots and Coefficients Quadratics ax 2  bx  c  0 c   a b    a Cubics ax 3  bx 2  cx  d  0 b       a c       a
  8. 8. Roots and Coefficients Quadratics ax 2  bx  c  0 c   a b    a Cubics ax 3  bx 2  cx  d  0 b       a d    a c       a
  9. 9. Roots and Coefficients Quadratics ax 2  bx  c  0 c   a b    a Cubics ax 3  bx 2  cx  d  0 b       a Quartics d    a c       a ax 4  bx 3  cx 2  dx  e  0
  10. 10. Roots and Coefficients Quadratics ax 2  bx  c  0 c   a b    a Cubics ax 3  bx 2  cx  d  0 b       a Quartics d    a c       a ax 4  bx 3  cx 2  dx  e  0 b        a
  11. 11. Roots and Coefficients Quadratics ax 2  bx  c  0 c   a b    a Cubics ax 3  bx 2  cx  d  0 b       a Quartics d    a c       a ax 4  bx 3  cx 2  dx  e  0 b        a c             a
  12. 12. Roots and Coefficients Quadratics ax 2  bx  c  0 c   a b    a Cubics ax 3  bx 2  cx  d  0 b       a Quartics d    a c       a ax 4  bx 3  cx 2  dx  e  0 c b                    a a d          a
  13. 13. Roots and Coefficients Quadratics ax 2  bx  c  0 c   a b    a Cubics ax 3  bx 2  cx  d  0 b       a Quartics d    a c       a ax 4  bx 3  cx 2  dx  e  0 c b                    a a d e            a a
  14. 14. For the polynomial equation; ax n  bx n1  cx n2  dx n3    0
  15. 15. For the polynomial equation; ax n  bx n1  cx n2  dx n3    0 b    a (sum of roots, one at a time)
  16. 16. For the polynomial equation; ax n  bx n1  cx n2  dx n3    0 b    a c   a (sum of roots, one at a time) (sum of roots, two at a time)
  17. 17. For the polynomial equation; ax n  bx n1  cx n2  dx n3    0 b    a c   a d    a (sum of roots, one at a time) (sum of roots, two at a time) (sum of roots, three at a time)
  18. 18. For the polynomial equation; ax n  bx n1  cx n2  dx n3    0     b  a c  a d  a e  a (sum of roots, one at a time) (sum of roots, two at a time) (sum of roots, three at a time) (sum of roots, four at a time)
  19. 19. For the polynomial equation; ax n  bx n1  cx n2  dx n3    0     b  a c  a d  a e  a Note:  (sum of roots, one at a time) (sum of roots, two at a time) (sum of roots, three at a time) (sum of roots, four at a time) 2      2  2
  20. 20. e.g. (i ) If  ,  and  are the roots of 2 x 3  5 x 2  3 x  1  0, find the values of; a) 4  4   4  7
  21. 21. e.g. (i ) If  ,  and  are the roots of 2 x 3  5 x 2  3 x  1  0, find the values of; a) 4  4   4  7 5      2
  22. 22. e.g. (i ) If  ,  and  are the roots of 2 x 3  5 x 2  3 x  1  0, find the values of; a) 4  4   4  7 5      2        3 2
  23. 23. e.g. (i ) If  ,  and  are the roots of 2 x 3  5 x 2  3 x  1  0, find the values of; a) 4  4   4  7 5      2        3 2    1 2
  24. 24. e.g. (i ) If  ,  and  are the roots of 2 x 3  5 x 2  3 x  1  0, find the values of; a) 4  4   4  7 5      2        5  1 4  4   4  7  4    7    2  2 3 2    1 2
  25. 25. e.g. (i ) If  ,  and  are the roots of 2 x 3  5 x 2  3 x  1  0, find the values of; a) 4  4   4  7 5      2        5  1 4  4   4  7  4    7    2  2 27  2 3 2    1 2
  26. 26. e.g. (i ) If  ,  and  are the roots of 2 x 3  5 x 2  3 x  1  0, find the values of; a) 4  4   4  7 5      2        5  1 4  4   4  7  4    7    2  2 27  2 1 1 1 b)      3 2    1 2
  27. 27. e.g. (i ) If  ,  and  are the roots of 2 x 3  5 x 2  3 x  1  0, find the values of; a) 4  4   4  7 5      2        5  1 4  4   4  7  4    7    2  2 27  2 1 1 1      b)        3 2    1 2
  28. 28. e.g. (i ) If  ,  and  are the roots of 2 x 3  5 x 2  3 x  1  0, find the values of; a) 4  4   4  7 5      2        5  1 4  4   4  7  4    7    2  2 27  2 1 1 1      b)        3  2 1  2  3 2    1 2
  29. 29. e.g. (i ) If  ,  and  are the roots of 2 x 3  5 x 2  3 x  1  0, find the values of; a) 4  4   4  7 5      2        5  1 4  4   4  7  4    7    2  2 27  2 1 1 1      b)        3  2 1  2  3  3 2    1 2
  30. 30. e.g. (i ) If  ,  and  are the roots of 2 x 3  5 x 2  3 x  1  0, find the values of; a) 4  4   4  7 5      2        3 2    5  1 4  4   4  7  4    7    2  2 27  2 1 1 1      b)    c)  2   2   2     3  2 1  2  3  1 2
  31. 31. e.g. (i ) If  ,  and  are the roots of 2 x 3  5 x 2  3 x  1  0, find the values of; a) 4  4   4  7 5      2        3 2    1 2 5  1 4  4   4  7  4    7    2  2 27  2 1 1 1      b)    c)  2   2   2    2          2       3   2 1  2  3
  32. 32. e.g. (i ) If  ,  and  are the roots of 2 x 3  5 x 2  3 x  1  0, find the values of; a) 4  4   4  7 5      2        3 2    1 2 5  1 4  4   4  7  4    7    2  2 27  2 1 1 1      b)    c)  2   2   2    2          2       3 2  5   3     2    2 2  2 1  2  3
  33. 33. e.g. (i ) If  ,  and  are the roots of 2 x 3  5 x 2  3 x  1  0, find the values of; a) 4  4   4  7 5      2        3 2    1 2 5  1 4  4   4  7  4    7    2  2 27  2 1 1 1      b)    c)  2   2   2    2          2       3 2  5   3     2    2 2  2 1  37 2  4  3
  34. 34. 1988 Extension 1 HSC Q2c) 3 If  ,  and  are the roots of x  3 x  1  0 find: (i)     
  35. 35. 1988 Extension 1 HSC Q2c) 3 If  ,  and  are the roots of x  3 x  1  0 find: (i)           0
  36. 36. 1988 Extension 1 HSC Q2c) 3 If  ,  and  are the roots of x  3 x  1  0 find: (i)           0 (ii) 
  37. 37. 1988 Extension 1 HSC Q2c) 3 If  ,  and  are the roots of x  3 x  1  0 find: (i)           0 (ii)    1
  38. 38. 1988 Extension 1 HSC Q2c) 3 If  ,  and  are the roots of x  3 x  1  0 find: (i)           0 (ii)    1 (iii) 1   1   1 
  39. 39. 1988 Extension 1 HSC Q2c) 3 If  ,  and  are the roots of x  3 x  1  0 find: (i)           0 (ii)    1 (iii) 1   1   1              1 1 1
  40. 40. 1988 Extension 1 HSC Q2c) 3 If  ,  and  are the roots of x  3 x  1  0 find: (i)           0 (ii)    1 (iii) 1   1   1              1 1 1  3 1
  41. 41. 1988 Extension 1 HSC Q2c) 3 If  ,  and  are the roots of x  3 x  1  0 find: (i)           0 (ii)    1 (iii) 1   1   1              1 1 1 3 1 3 
  42. 42. 2003 Extension 1 HSC Q4c) It is known that two of the roots of the equation 2 x 3  x 2  kx  6  0 are reciprocals of each other. Find the value of k.
  43. 43. 2003 Extension 1 HSC Q4c) It is known that two of the roots of the equation 2 x 3  x 2  kx  6  0 are reciprocals of each other. Find the value of k. 1 Let the roots be  , and  
  44. 44. 2003 Extension 1 HSC Q4c) It is known that two of the roots of the equation 2 x 3  x 2  kx  6  0 are reciprocals of each other. Find the value of k. 1 Let the roots be  , and   1      6    2   
  45. 45. 2003 Extension 1 HSC Q4c) It is known that two of the roots of the equation 2 x 3  x 2  kx  6  0 are reciprocals of each other. Find the value of k. 1 Let the roots be  , and   1      6    2     3 
  46. 46. 2003 Extension 1 HSC Q4c) It is known that two of the roots of the equation 2 x 3  x 2  kx  6  0 are reciprocals of each other. Find the value of k. 1 Let the roots be  , and   1      6    2     3  P   3  0
  47. 47. 2003 Extension 1 HSC Q4c) It is known that two of the roots of the equation 2 x 3  x 2  kx  6  0 are reciprocals of each other. Find the value of k. 1 Let the roots be  , and   1      6    2     3  P   3  0 2 3   3  k  3  6  0 3 2
  48. 48. 2003 Extension 1 HSC Q4c) It is known that two of the roots of the equation 2 x 3  x 2  kx  6  0 are reciprocals of each other. Find the value of k. 1 Let the roots be  , and   1      6    2     3  P   3  0 2 3   3  k  3  6  0 3 2  54  9  3k  6  0
  49. 49. 2003 Extension 1 HSC Q4c) It is known that two of the roots of the equation 2 x 3  x 2  kx  6  0 are reciprocals of each other. Find the value of k. 1 Let the roots be  , and   1      6    2     3  P   3  0 2 3   3  k  3  6  0 3 2  54  9  3k  6  0 3k  39
  50. 50. 2003 Extension 1 HSC Q4c) It is known that two of the roots of the equation 2 x 3  x 2  kx  6  0 are reciprocals of each other. Find the value of k. 1 Let the roots be  , and   1      6    2     3  P   3  0 2 3   3  k  3  6  0 3 2  54  9  3k  6  0 3k  39 k  13
  51. 51. 2006 Extension 1 HSC Q4a) The cubic polynomial P x   x 3  rx 2  sx  t , where r, s and t are real numbers, has three real zeros, 1,  and   (i) Find the value of r
  52. 52. 2006 Extension 1 HSC Q4a) The cubic polynomial P x   x 3  rx 2  sx  t , where r, s and t are real numbers, has three real zeros, 1,  and   (i) Find the value of r 1       r
  53. 53. 2006 Extension 1 HSC Q4a) The cubic polynomial P x   x 3  rx 2  sx  t , where r, s and t are real numbers, has three real zeros, 1,  and   (i) Find the value of r 1       r r  1
  54. 54. 2006 Extension 1 HSC Q4a) The cubic polynomial P x   x 3  rx 2  sx  t , where r, s and t are real numbers, has three real zeros, 1,  and   (i) Find the value of r 1       r r  1 (ii) Find the value of s + t
  55. 55. 2006 Extension 1 HSC Q4a) The cubic polynomial P x   x 3  rx 2  sx  t , where r, s and t are real numbers, has three real zeros, 1,  and   (i) Find the value of r 1       r r  1 (ii) Find the value of s + t 1   1         s
  56. 56. 2006 Extension 1 HSC Q4a) The cubic polynomial P x   x 3  rx 2  sx  t , where r, s and t are real numbers, has three real zeros, 1,  and   (i) Find the value of r 1       r r  1 (ii) Find the value of s + t 1   1         s s   2
  57. 57. 2006 Extension 1 HSC Q4a) The cubic polynomial P x   x 3  rx 2  sx  t , where r, s and t are real numbers, has three real zeros, 1,  and   (i) Find the value of r 1       r r  1 (ii) Find the value of s + t 1   1         s s   2 1     t
  58. 58. 2006 Extension 1 HSC Q4a) The cubic polynomial P x   x 3  rx 2  sx  t , where r, s and t are real numbers, has three real zeros, 1,  and   (i) Find the value of r 1       r r  1 (ii) Find the value of s + t 1   1         s s   2 1     t t 2
  59. 59. 2006 Extension 1 HSC Q4a) The cubic polynomial P x   x 3  rx 2  sx  t , where r, s and t are real numbers, has three real zeros, 1,  and   (i) Find the value of r 1       r r  1 (ii) Find the value of s + t 1   1         s s   2 1     t t 2 s  t  0
  60. 60. Exercise 4F; 2, 4, 5ac, 6ac, 8, 10a, 13, 15, 16ad, 17, 18, 19

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