The document discusses relationships between the coefficients and roots of polynomials. It states that for a polynomial P(x) = ax^n + bx^(n-1) + cx^(n-2) + ..., the sum of the roots is equal to -b/a, the sum of the roots taken two at a time is equal to c/a, and so on for higher order terms. It also generalizes relationships between symmetric polynomials of the roots. For example, it states the sum of the squares of the roots can be expressed in terms of the sum of the roots and the sum of the roots taken two at a time. The document provides an example polynomial and calculates its root-related coefficients.
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
a
(sum of roots, 2 at a time i.e. )
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
a
(sum of roots, 2 at a time i.e. )
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
a
(sum of roots, 2 at a time i.e. )
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 2
1
(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)
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)
Substitute (1) into (3) k 2a c
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 c
2 d ( k 2 )
a a 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 c
2 d ( k 2 )
a a a
abc c 2 a 2 d
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 c
2 d ( k 2 )
a a a
abc c 2 a 2 d
c 2 a 2 d abc
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 2ki 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 2ki ki ki 2 b
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 2ki ki ki 2 b
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 2ki ki ki 2 b
k 2 2 0
k 2 2
From c) c k 2 a
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 2ki ki ki 2 b
k 2 2 0
k 2 2
From c) c k 2 a
c 2a
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 2ki ki ki 2 b
k 2 2 0
k 2 2
From c) c k 2 a
c 2a
c 2M
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 2ki ki ki 2 b
k 2 2 0
k 2 2
From c) c k 2 a
c 2a
c 2 M and a are both integers, M is an integer
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 2ki ki ki 2 b
k 2 2 0
k 2 2
From c) c k 2 a
c 2a
c 2 M and a are both integers, M is an integer
Thus c is an even number, as it is divisible by 2