1. Roots and Coefficients

2. Roots and Coefficients
For P x   ax n  bx n 1  cx n  2 

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. 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. 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. 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. We have already seen that for ax 2  bx  c;
 2   2     2  2

8. We have already seen that for ax 2  bx  c;
 2   2     2  2
and
1 1 
 




9. We have already seen that for ax 2  bx  c;
 2   2     2  2
and
1 1 
 


This can be generalised to;


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. 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. 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. 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. e.g. (i) x 3  2 x 2  3 x  4  0, find;

15. e.g. (i) x 3  2 x 2  3 x  4  0, find;
a)  
b)   
c)   

16. e.g. (i) x 3  2 x 2  3 x  4  0, find;
a)  
b)   
c)   
=2
=3
=4

17. e.g. (i) x 3  2 x 2  3 x  4  0, find;
a)  
b)   
c)   
d)   2
=2
=3
=4

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. 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. 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. 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. 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. 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.  3  2  2  3   12  0


25.  3  2  2  3   12  0

 3  2  2  3   12


26.  3  2  2  3   12  0

 3  2  2  3   12

 2 2   32   12
2

27.  3  2  2  3   12  0

 3  2  2  3   12

 2 2   32   12
2
f)   4

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.  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.  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.  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.  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. g)           

34. g)           
             

35. g)           
             
 2   2   2   

36. g)           
             
 2   2   2   
 4  2  2    2   
 8  4  4   2   4  2  2  
 8  4   2    

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. (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. (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. (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. (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. (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. (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. (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. (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. (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. c) Show that c 2  a 2 d  abc

48. c) Show that c 2  a 2 d  abc
Using (2); q  b  k 2

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. 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. 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. 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. 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. 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. d) If 2 is also a root of the equation, and b = 0, show that c is even.

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. 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. 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. 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. 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. 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. 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. 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. 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. 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