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# X2 t02 03 roots & coefficients (2013)

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### X2 t02 03 roots & coefficients (2013)

1. 1. Roots and Coefficients
2. 2. Roots and Coefficients For P x   ax n  bx n 1  cx n  2 
3. 3. Roots and Coefficients For P x   ax n  bx n 1  cx n  2  b   a (sum of roots, 1 at a time i.e.        )
4. 4. Roots and Coefficients For P x   ax n  bx n 1  cx n  2  b   a (sum of roots, 1 at a time i.e.        ) c   (sum of roots, 2 at a time i.e.       )  a
5. 5. Roots and Coefficients For P x   ax n  bx n 1  cx n  2  b   a (sum of roots, 1 at a time i.e.        ) c   (sum of roots, 2 at a time i.e.       )  a d   a (sum of roots, 3 at a time i.e.    )
6. 6. Roots and Coefficients For P x   ax n  bx n 1  cx n  2  b   a (sum of roots, 1 at a time i.e.        ) c   (sum of roots, 2 at a time i.e.       )  a d   a (sum of roots, 3 at a time i.e.    ) e   a (sum of roots, 4 at a time)
7. 7. We have already seen that for ax 2  bx  c;  2   2     2  2
8. 8. We have already seen that for ax 2  bx  c;  2   2     2  2 and 1 1      
9. 9. We have already seen that for ax 2  bx  c;  2   2     2  2 and 1 1      This can be generalised to; 
10. 10. We have already seen that for ax 2  bx  c;  2   2     2  2 and 1 1       This can be generalised to;  2      2  2
11. 11. We have already seen that for ax 2  bx  c;  2   2     2  2 and 1 1       This can be generalised to;  2      2       1 2 (order 2)
12. 12. We have already seen that for ax 2  bx  c;  2   2     2  2 and 1 1       This can be generalised to;  2      2  2      1    (order 2) (order 3)
13. 13. We have already seen that for ax 2  bx  c;  2   2     2  2 and 1 1       This can be generalised to;       2  2 2      1       (order 2) (order 3) (order 4)
14. 14. e.g. (i) x 3  2 x 2  3 x  4  0, find;
15. 15. e.g. (i) x 3  2 x 2  3 x  4  0, find; a)   b)    c)   
16. 16. e.g. (i) x 3  2 x 2  3 x  4  0, find; a)   b)    c)    =2 =3 =4
17. 17. e.g. (i) x 3  2 x 2  3 x  4  0, find; a)   b)    c)    d)   2 =2 =3 =4
18. 18. e.g. (i) x 3  2 x 2  3 x  4  0, find; a)   b)    c)    d)   2 =2 =3 =4      2  2
19. 19. e.g. (i) x 3  2 x 2  3 x  4  0, find; a)   b)    c)    d)   2 =2 =3 =4      2  2  2 2  23  2
20. 20. e.g. (i) x 3  2 x 2  3 x  4  0, find; a)   b)    c)    d)   2 =2 =3 =4      2  2  2 2  23  2 e)   3
21. 21. e.g. (i) x 3  2 x 2  3 x  4  0, find; a)   b)    c)    d)   2 =2 =3 =4      2  2  2 2  23  2 e)   3  3  2 2  3  4  0
22. 22. e.g. (i) x 3  2 x 2  3 x  4  0, find; a)   b)    c)    d)   2 =2 =3 =4      2  2  2 2  23  2 e)   3  3  2 2  3  4  0  3  2  2  3  4  0  3  2 2  3  4  0
23. 23. e.g. (i) x 3  2 x 2  3 x  4  0, find; a)   b)    c)    d)   2 =2 =3 =4      2  2  2 2  23  2 e)   3  3  2 2  3  4  0  3  2  2  3  4  0  3  2 2  3  4  0  3  2  2  3   12  0 
24. 24.  3  2  2  3   12  0 
25. 25.  3  2  2  3   12  0   3  2  2  3   12 
26. 26.  3  2  2  3   12  0   3  2  2  3   12   2 2   32   12 2
27. 27.  3  2  2  3   12  0   3  2  2  3   12   2 2   32   12 2 f)   4
28. 28.  3  2  2  3   12  0   3  2  2  3   12   2 2   32   12 2 f)   4  4  2 3  3 2  4  0
29. 29.  3  2  2  3   12  0   3  2  2  3   12   2 2   32   12 2 f)   4  4  2 3  3 2  4  0  4  2  3  3 2  4   0  4  2 3  3 2  4  0
30. 30.  3  2  2  3   12  0   3  2  2  3   12   2 2   32   12 2 f)   4  4  2 3  3 2  4  0  4  2  3  3 2  4   0  4  2 3  3 2  4  0  4  2  3  3  2  4   0 
31. 31.  3  2  2  3   12  0   3  2  2  3   12   2 2   32   12 2 f)   4  4  2 3  3 2  4  0  4  2  3  3 2  4   0  4  2 3  3 2  4  0  4  2  3  3  2  4   0   4  2  3  3  2  4  
32. 32.  3  2  2  3   12  0   3  2  2  3   12   2 2   32   12 2 f)   4  4  2 3  3 2  4  0  4  2  3  3 2  4   0  4  2 3  3 2  4  0  4  2  3  3  2  4   0   4  2  3  3  2  4    22   3 2   42   18
33. 33. g)           
34. 34. g)                         
35. 35. g)                           2   2   2   
36. 36. g)                           2   2   2     4  2  2    2     8  4  4   2   4  2  2    8  4   2    
37. 37. g)                           2   2   2     4  2  2    2     8  4  4   2   4  2  2    8  4   2      8  42   23  4 2
38. 38. (ii) (1996) Consider the polynomial equation x 4  ax 3  bx 2  cx  d  0 where a, b, c and d are all integers. Suppose the equation has a root of the form ki, where k is real and k  0
39. 39. (ii) (1996) Consider the polynomial equation x 4  ax 3  bx 2  cx  d  0 where a, b, c and d are all integers. Suppose the equation has a root of the form ki, where k is real and k  0 a) State why the conjugate –ki is also a root.
40. 40. (ii) (1996) Consider the polynomial equation x 4  ax 3  bx 2  cx  d  0 where a, b, c and d are all integers. Suppose the equation has a root of the form ki, where k is real and k  0 a) State why the conjugate –ki is also a root. Roots appear in conjugate pairs when the coefficients are real.
41. 41. (ii) (1996) Consider the polynomial equation x 4  ax 3  bx 2  cx  d  0 where a, b, c and d are all integers. Suppose the equation has a root of the form ki, where k is real and k  0 a) State why the conjugate –ki is also a root. Roots appear in conjugate pairs when the coefficients are real. b) Show that c  k 2 a
42. 42. (ii) (1996) Consider the polynomial equation x 4  ax 3  bx 2  cx  d  0 where a, b, c and d are all integers. Suppose the equation has a root of the form ki, where k is real and k  0 a) State why the conjugate –ki is also a root. Roots appear in conjugate pairs when the coefficients are real. b) Show that c  k 2 a x 4  ax 3  bx 2  cx  d   x  ki  x  ki x 2  px  q 
43. 43. (ii) (1996) Consider the polynomial equation x 4  ax 3  bx 2  cx  d  0 where a, b, c and d are all integers. Suppose the equation has a root of the form ki, where k is real and k  0 a) State why the conjugate –ki is also a root. Roots appear in conjugate pairs when the coefficients are real. b) Show that c  k 2 a x 4  ax 3  bx 2  cx  d   x  ki  x  ki x 2  px  q   x 2  k 2 x 2  px  q   x 4  px 3  qx 2  k 2 x 2  k 2 px  k 2 q  x 4  px 3  q  k 2 x 2  k 2 px  k 2 q
44. 44. (ii) (1996) Consider the polynomial equation x 4  ax 3  bx 2  cx  d  0 where a, b, c and d are all integers. Suppose the equation has a root of the form ki, where k is real and k  0 a) State why the conjugate –ki is also a root. Roots appear in conjugate pairs when the coefficients are real. b) Show that c  k 2 a x 4  ax 3  bx 2  cx  d   x  ki  x  ki x 2  px  q   x 2  k 2 x 2  px  q   x 4  px 3  qx 2  k 2 x 2  k 2 px  k 2 q  x 4  px 3  q  k 2 x 2  k 2 px  k 2 q  p  a  (1) q  k 2  b  (2) k 2 p  c  (3) k 2 q  d  (4)
45. 45. (ii) (1996) Consider the polynomial equation x 4  ax 3  bx 2  cx  d  0 where a, b, c and d are all integers. Suppose the equation has a root of the form ki, where k is real and k  0 a) State why the conjugate –ki is also a root. Roots appear in conjugate pairs when the coefficients are real. b) Show that c  k 2 a x 4  ax 3  bx 2  cx  d   x  ki  x  ki x 2  px  q   x 2  k 2 x 2  px  q   x 4  px 3  qx 2  k 2 x 2  k 2 px  k 2 q  x 4  px 3  q  k 2 x 2  k 2 px  k 2 q  p  a  (1) q  k 2  b  (2) k 2 p  c  (3) k 2 q  d  (4) Substitute (1) into (3)
46. 46. (ii) (1996) Consider the polynomial equation x 4  ax 3  bx 2  cx  d  0 where a, b, c and d are all integers. Suppose the equation has a root of the form ki, where k is real and k  0 a) State why the conjugate –ki is also a root. Roots appear in conjugate pairs when the coefficients are real. b) Show that c  k 2 a x 4  ax 3  bx 2  cx  d   x  ki  x  ki x 2  px  q   x 2  k 2 x 2  px  q   x 4  px 3  qx 2  k 2 x 2  k 2 px  k 2 q  x 4  px 3  q  k 2 x 2  k 2 px  k 2 q  p  a  (1) q  k 2  b  (2) k 2 p  c  (3) k 2 q  d  (4) k 2a  c Substitute (1) into (3)
47. 47. c) Show that c 2  a 2 d  abc
48. 48. c) Show that c 2  a 2 d  abc Using (2); q  b  k 2
49. 49. c) Show that c 2  a 2 d  abc Using (2); q  b  k 2 Sub into (4); k 2 b  k 2   d
50. 50. c) Show that c 2  a 2 d  abc Using (2); q  b  k 2 Sub into (4); k 2 b  k 2   d bk 2  k 4  d
51. 51. c) Show that c 2  a 2 d  abc Using (2); q  b  k 2 Sub into (4); k 2 b  k 2   d bk 2  k 4  d bc c 2  2 d a a
52. 52. c) Show that c 2  a 2 d  abc Using (2); q  b  k 2 Sub into (4); k 2 b  k 2   d bk 2  k 4  d bc c 2  2 d a a c ( k 2  ) a
53. 53. c) Show that c 2  a 2 d  abc Using (2); q  b  k 2 Sub into (4); k 2 b  k 2   d bk 2  k 4  d bc c 2  2 d a a abc  c 2  a 2 d c ( k 2  ) a
54. 54. c) Show that c 2  a 2 d  abc Using (2); q  b  k 2 Sub into (4); k 2 b  k 2   d bk 2  k 4  d bc c 2  2 d a a abc  c 2  a 2 d c 2  a 2 d  abc c ( k 2  ) a
55. 55. d) If 2 is also a root of the equation, and b = 0, show that c is even.
56. 56. d) If 2 is also a root of the equation, and b = 0, show that c is even. Let  be the 4th root
57. 57. d) If 2 is also a root of the equation, and b = 0, show that c is even. Let  be the 4th root As the other three roots are integer multiples, and the product of the roots is an integer;  is an integer
58. 58. d) If 2 is also a root of the equation, and b = 0, show that c is even. Let  be the 4th root As the other three roots are integer multiples, and the product of the roots is an integer;  is an integer  ki ki   2 ki   2ki     ki    ki   2  b   
59. 59. d) If 2 is also a root of the equation, and b = 0, show that c is even. Let  be the 4th root As the other three roots are integer multiples, and the product of the roots is an integer;  is an integer  ki ki   2 ki   2ki     ki    ki   2  b k 2  2  0 k 2  2   
60. 60. d) If 2 is also a root of the equation, and b = 0, show that c is even. Let  be the 4th root As the other three roots are integer multiples, and the product of the roots is an integer;  is an integer  ki ki   2 ki   2ki     ki    ki   2  b From c)  c  k 2 a k 2  2  0 k 2  2   
61. 61. d) If 2 is also a root of the equation, and b = 0, show that c is even. Let  be the 4th root As the other three roots are integer multiples, and the product of the roots is an integer;  is an integer  ki ki   2 ki   2ki     ki    ki   2  b From c)  c  k 2 a c  2a k 2  2  0 k 2  2   
62. 62. d) If 2 is also a root of the equation, and b = 0, show that c is even. Let  be the 4th root As the other three roots are integer multiples, and the product of the roots is an integer;  is an integer  ki ki   2 ki   2ki     ki    ki   2  b From c)  c  k 2 a c  2a c  2M k 2  2  0 k 2  2   
63. 63. d) If 2 is also a root of the equation, and b = 0, show that c is even. Let  be the 4th root As the other three roots are integer multiples, and the product of the roots is an integer;  is an integer  ki ki   2 ki   2ki     ki    ki   2  b From c)  c  k 2 a    k 2  2  0 k 2  2 c  2a c  2 M  and a are both integers, M is an integer
64. 64. d) If 2 is also a root of the equation, and b = 0, show that c is even. Let  be the 4th root As the other three roots are integer multiples, and the product of the roots is an integer;  is an integer  ki ki   2 ki   2ki     ki    ki   2  b From c)  c  k 2 a c  2a c  2M    k 2  2  0 k 2  2  and a are both integers, M is an integer Thus c is an even number, as it is divisible by 2
65. 65. d) If 2 is also a root of the equation, and b = 0, show that c is even. Let  be the 4th root As the other three roots are integer multiples, and the product of the roots is an integer;  is an integer  ki ki   2 ki   2ki     ki    ki   2  b From c)  c  k 2 a c  2a c  2M    k 2  2  0 k 2  2  and a are both integers, M is an integer Thus c is an even number, as it is divisible by 2 Patel: Exercise 5D; 1 to 9