2. Multiple Roots
If P(x), has a root, x = a, of multiplicity m,
then P’(x) has a root, x = a, of multiplicity m - 1
3. Multiple Roots
If P(x), has a root, x = a, of multiplicity m,
then P’(x) has a root, x = a, of multiplicity m - 1
Proof:
P x x a Q x
m
(m > 1, x = a is not a root of Q(x))
4. Multiple Roots
If P(x), has a root, x = a, of multiplicity m,
then P’(x) has a root, x = a, of multiplicity m - 1
Proof:
P x x a Q x
m
(m > 1, x = a is not a root of Q(x))
P x x a Q x Q x m x a
1
m 1
x a x a Q x mQ x
m
m 1
5. Multiple Roots
If P(x), has a root, x = a, of multiplicity m,
then P’(x) has a root, x = a, of multiplicity m - 1
Proof:
P x x a Q x
m
(m > 1, x = a is not a root of Q(x))
P x x a Q x Q x m x a
1
m 1
x a x a Q x mQ x
m1
x a R x
(where x = a is not a root of R(x))
m
m 1
6. Multiple Roots
If P(x), has a root, x = a, of multiplicity m,
then P’(x) has a root, x = a, of multiplicity m - 1
Proof:
P x x a Q x
m
(m > 1, x = a is not a root of Q(x))
P x x a Q x Q x m x a
1
m 1
x a x a Q x mQ x
m1
x a R x
(where x = a is not a root of R(x))
m
m 1
P’(x) has a root, x = a, of multiplicity m - 1
7. e.g. (i) Solve the equation x 3 4 x 2 3 x 18 0 , given that it has a
double root
8. e.g. (i) Solve the equation x 3 4 x 2 3 x 18 0 , given that it has a
double root
P x x 3 4 x 2 3 x 18
P x 3 x 2 8 x 3
9. e.g. (i) Solve the equation x 3 4 x 2 3 x 18 0 , given that it has a
double root
P x x 3 4 x 2 3 x 18
P x 3 x 2 8 x 3
3 x 1 x 3
1
x or x 3
double root is
3
10. e.g. (i) Solve the equation x 3 4 x 2 3 x 18 0 , given that it has a
double root
P x x 3 4 x 2 3 x 18
P x 3 x 2 8 x 3
3 x 1 x 3
1
x or x 3
double root is
3
NOT POSSIBLE
As (3x + 1) is not a factor
11. e.g. (i) Solve the equation x 3 4 x 2 3 x 18 0 , given that it has a
double root
P x x 3 4 x 2 3 x 18
P x 3 x 2 8 x 3
3 x 1 x 3
1
x or x 3
double root is
3
NOT POSSIBLE
As (3x + 1) is not a factor
x 3 4 x 2 3 x 18 0
x 32 x 2 0
12. e.g. (i) Solve the equation x 3 4 x 2 3 x 18 0 , given that it has a
double root
P x x 3 4 x 2 3 x 18
P x 3 x 2 8 x 3
3 x 1 x 3
1
x or x 3
double root is
3
NOT POSSIBLE
As (3x + 1) is not a factor
x 3 4 x 2 3 x 18 0
x 32 x 2 0
x 2 or x 3
13. (ii) (1991)
Let x be a root of the quartic polynomial;
P x x 4 Ax 3 Bx 2 Ax 1
where 2 B 2 4 A2
a) show that cannot be 0, 1 or -1
14. (ii) (1991)
Let x be a root of the quartic polynomial;
P x x 4 Ax 3 Bx 2 Ax 1
where 2 B 2 4 A2
a) show that cannot be 0, 1 or -1
P0 1 0, 0
15. (ii) (1991)
Let x be a root of the quartic polynomial;
P x x 4 Ax 3 Bx 2 Ax 1
where 2 B 2 4 A2
a) show that cannot be 0, 1 or -1
P0 1 0, 0
P1 1 A B A 1
2A B 2
16. (ii) (1991)
Let x be a root of the quartic polynomial;
P x x 4 Ax 3 Bx 2 Ax 1
where 2 B 2 4 A2
a) show that cannot be 0, 1 or -1
P0 1 0, 0
P1 1 A B A 1
2A B 2
P 1 1 A B A 1
2 A B 2
17. (ii) (1991)
Let x be a root of the quartic polynomial;
P x x 4 Ax 3 Bx 2 Ax 1
where 2 B 2 4 A2
a) show that cannot be 0, 1 or -1
P0 1 0, 0
P1 1 A B A 1
2A B 2
BUT
2 B 2 4 A2
P 1 1 A B A 1
2 A B 2
18. (ii) (1991)
Let x be a root of the quartic polynomial;
P x x 4 Ax 3 Bx 2 Ax 1
where 2 B 2 4 A2
a) show that cannot be 0, 1 or -1
P0 1 0, 0
P1 1 A B A 1
2A B 2
BUT
2 B 2 4 A2
2 B 2 A
2A B 2 0
P 1 1 A B A 1
2 A B 2
19. (ii) (1991)
Let x be a root of the quartic polynomial;
P x x 4 Ax 3 Bx 2 Ax 1
where 2 B 2 4 A2
a) show that cannot be 0, 1 or -1
P0 1 0, 0
P1 1 A B A 1
2A B 2
BUT
2 B 2 4 A2
2 B 2 A
2A B 2 0
P1 0, P 1 0
hence 1
P 1 1 A B A 1
2 A B 2
22. b) Show that
1
is a root
1 1 A B A 1
P
4
3
2
1 A B 2 A 3 4
4
23. b) Show that
1
is a root
1 1 A B A 1
P
4
3
2
1 A B 2 A 3 4
P
4
4
24. b) Show that
1
is a root
1 1 A B A 1
P
4
3
2
1 A B 2 A 3 4
P
4
4
0
4
0
( P 0 as is a root)
25. b) Show that
1
is a root
1 1 A B A 1
P
4
3
2
1 A B 2 A 3 4
P
4
4
0
4
0
1
is a root of P x
( P 0 as is a root)
26. c) Deduce that if is a multiple root, then its multiplicity is 2 and
4 B 8 A2
27. c) Deduce that if is a multiple root, then its multiplicity is 2 and
4 B 8 A2
1
If is a double root of P x , then so is , which accounts for 4 roots
28. c) Deduce that if is a multiple root, then its multiplicity is 2 and
4 B 8 A2
1
If is a double root of P x , then so is , which accounts for 4 roots
However P(x) is a quartic which has a maximum of 4 roots
29. c) Deduce that if is a multiple root, then its multiplicity is 2 and
4 B 8 A2
1
If is a double root of P x , then so is , which accounts for 4 roots
However P(x) is a quartic which has a maximum of 4 roots
Thus no roots can have a multiplicity > 2
30. c) Deduce that if is a multiple root, then its multiplicity is 2 and
4 B 8 A2
1
If is a double root of P x , then so is , which accounts for 4 roots
However P(x) is a quartic which has a maximum of 4 roots
Thus no roots can have a multiplicity > 2
P x 4 x 3 Ax 2 Bx A
3
2
let the roots be ,
1
and
31. c) Deduce that if is a multiple root, then its multiplicity is 2 and
4 B 8 A2
1
If is a double root of P x , then so is , which accounts for 4 roots
However P(x) is a quartic which has a maximum of 4 roots
Thus no roots can have a multiplicity > 2
P x 4 x 3 Ax 2 Bx A
1
3
A
4
1
1 B
2
1
A
4
3
2
let the roots be ,
1
sum of roots (1)
(2)
(3)
and