X2 T02 01 multiple roots
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Transcript of X2 T02 01 multiple roots
Polynomials
PolynomialsIf (x – a) is a factor of P(x), then P(a) = 0
PolynomialsIf (x – a) is a factor of P(x), then P(a) = 0
If (ax – b) is a factor of P(x), then = 0
PolynomialsIf (x – a) is a factor of P(x), then P(a) = 0
abPIf (ax – b) is a factor of P(x), then = 0
Multiple Roots
Multiple RootsIf P(x), has a root, x = a, of multiplicity m,
then P’(x) has a root, x = a, of multiplicity m - 1
Multiple RootsIf P(x), has a root, x = a, of multiplicity m,
then P’(x) has a root, x = a, of multiplicity m - 1
Proof: xQaxxP m
Multiple RootsIf P(x), has a root, x = a, of multiplicity m,
then P’(x) has a root, x = a, of multiplicity m - 1
Proof: xQaxxP m (m > 1, x = a is not a root of Q(x))
Multiple RootsIf P(x), has a root, x = a, of multiplicity m,
then P’(x) has a root, x = a, of multiplicity m - 1
Proof: xQaxxP m
11 mm axmxQxQaxxP
(m > 1, x = a is not a root of Q(x))
Multiple RootsIf P(x), has a root, x = a, of multiplicity m,
then P’(x) has a root, x = a, of multiplicity m - 1
Proof: xQaxxP m
11 mm axmxQxQaxxP
xmQxQaxax m 1
(m > 1, x = a is not a root of Q(x))
Multiple RootsIf P(x), has a root, x = a, of multiplicity m,
then P’(x) has a root, x = a, of multiplicity m - 1
Proof: xQaxxP m
11 mm axmxQxQaxxP
xmQxQaxax m 1
xRax m 1
(m > 1, x = a is not a root of Q(x))
Multiple RootsIf P(x), has a root, x = a, of multiplicity m,
then P’(x) has a root, x = a, of multiplicity m - 1
Proof: xQaxxP m
11 mm axmxQxQaxxP
xmQxQaxax m 1
xRax m 1
(m > 1, x = a is not a root of Q(x))
(where x = a is not a root of R(x))
Multiple RootsIf P(x), has a root, x = a, of multiplicity m,
then P’(x) has a root, x = a, of multiplicity m - 1
Proof: xQaxxP m
11 mm axmxQxQaxxP
xmQxQaxax m 1
xRax m 1
(m > 1, x = a is not a root of Q(x))
(where x = a is not a root of R(x))
P’(x) has a root, x = a, of multiplicity m - 1
e.g. (i) Solve the equation , given that it has a double root
01834 23 xxx
e.g. (i) Solve the equation , given that it has a double root
01834 23 xxx
1834 23 xxxxP
e.g. (i) Solve the equation , given that it has a double root
01834 23 xxx
1834 23 xxxxP 383 2 xxxP
e.g. (i) Solve the equation , given that it has a double root
01834 23 xxx
1834 23 xxxxP 383 2 xxxP
313 xx
e.g. (i) Solve the equation , given that it has a double root
01834 23 xxx
1834 23 xxxxP 383 2 xxxP
313 xx
double root is or 13
x 3x
e.g. (i) Solve the equation , given that it has a double root
01834 23 xxx
1834 23 xxxxP 383 2 xxxP
313 xx
double root is or
NOT POSSIBLEAs (3x + 1) is not a factor
13
x 3x
e.g. (i) Solve the equation , given that it has a double root
01834 23 xxx
1834 23 xxxxP 383 2 xxxP
313 xx
double root is or
NOT POSSIBLEAs (3x + 1) is not a factor
01834 23 xxx
13
x 3x
e.g. (i) Solve the equation , given that it has a double root
01834 23 xxx
1834 23 xxxxP 383 2 xxxP
313 xx
double root is or
NOT POSSIBLEAs (3x + 1) is not a factor
023 2 xx01834 23 xxx
13
x 3x
e.g. (i) Solve the equation , given that it has a double root
01834 23 xxx
1834 23 xxxxP 383 2 xxxP
313 xx
double root is or
NOT POSSIBLEAs (3x + 1) is not a factor
023 2 xx01834 23 xxx
3or 2 xx
13
x 3x
(ii) (1991)Let be a root of the quartic polynomial;
where
x 1234 AxBxAxxxP
22 42 AB
a) show that cannot be 0, 1 or -1
(ii) (1991)Let be a root of the quartic polynomial;
where
x 1234 AxBxAxxxP
22 42 AB
a) show that cannot be 0, 1 or -1 0 ,010 P
(ii) (1991)Let be a root of the quartic polynomial;
where
x 1234 AxBxAxxxP
22 42 AB
a) show that cannot be 0, 1 or -1 0 ,010 P
22
111
BA
ABAP
(ii) (1991)Let be a root of the quartic polynomial;
where
x 1234 AxBxAxxxP
22 42 AB
a) show that cannot be 0, 1 or -1 0 ,010 P
22
111
BA
ABAP 22
111
BA
ABAP
(ii) (1991)Let be a root of the quartic polynomial;
where
x 1234 AxBxAxxxP
22 42 AB
a) show that cannot be 0, 1 or -1 0 ,010 P
22
111
BA
ABAP 22
111
BA
ABAP
BUT 22 42 AB
(ii) (1991)Let be a root of the quartic polynomial;
where
x 1234 AxBxAxxxP
22 42 AB
a) show that cannot be 0, 1 or -1 0 ,010 P
22
111
BA
ABAP 22
111
BA
ABAP
BUT 22 42 AB
AB 22
(ii) (1991)Let be a root of the quartic polynomial;
where
x 1234 AxBxAxxxP
22 42 AB
a) show that cannot be 0, 1 or -1 0 ,010 P
22
111
BA
ABAP 22
111
BA
ABAP
BUT 22 42 AB
AB 22 022 BA
(ii) (1991)Let be a root of the quartic polynomial;
where
x 1234 AxBxAxxxP
22 42 AB
a) show that cannot be 0, 1 or -1 0 ,010 P
22
111
BA
ABAP 22
111
BA
ABAP
BUT 22 42 AB
AB 22 022 BA
01 ,01 PP
(ii) (1991)Let be a root of the quartic polynomial;
where
x 1234 AxBxAxxxP
22 42 AB
a) show that cannot be 0, 1 or -1 0 ,010 P
22
111
BA
ABAP 22
111
BA
ABAP
BUT 22 42 AB
AB 22 022 BA
01 ,01 PP1 hence
b) Show that is a root1
b) Show that is a root1
111234
ABAP
b) Show that is a root1
111234
ABAP
4
4321
ABA
b) Show that is a root1
111234
ABAP
4
4321
ABA
4P
b) Show that is a root1
111234
ABAP
4
4321
ABA
4P
40
b) Show that is a root1
111234
ABAP
4
4321
ABA
4P
40
root)aisas 0( P
b) Show that is a root1
111234
ABAP
4
4321
ABA
4P
40
root)aisas 0( P
0
b) Show that is a root1
111234
ABAP
4
4321
ABA
4P
40
root)aisas 0( P
0
xP ofroot a is 1
c) Deduce that if is a multiple root, then its multiplicity is 2 and284 AB
c) Deduce that if is a multiple root, then its multiplicity is 2 and284 AB
roots 4for accounts which ,1 is so then , ofroot double a is If
xP
c) Deduce that if is a multiple root, then its multiplicity is 2 and284 AB
roots 4for accounts which ,1 is so then , ofroot double a is If
xP
However P(x) is a quartic which has a maximum of 4 roots
c) Deduce that if is a multiple root, then its multiplicity is 2 and284 AB
roots 4for accounts which ,1 is so then , ofroot double a is If
xP
However P(x) is a quartic which has a maximum of 4 roots
Thus no roots can have a multiplicity > 2
c) Deduce that if is a multiple root, then its multiplicity is 2 and284 AB
roots 4for accounts which ,1 is so then , ofroot double a is If
xP
However P(x) is a quartic which has a maximum of 4 roots
Thus no roots can have a multiplicity > 2
ABxAxxxP 234 23
c) Deduce that if is a multiple root, then its multiplicity is 2 and284 AB
roots 4for accounts which ,1 is so then , ofroot double a is If
xP
However P(x) is a quartic which has a maximum of 4 roots
Thus no roots can have a multiplicity > 2
ABxAxxxP 234 23
and 1 , be roots let the
c) Deduce that if is a multiple root, then its multiplicity is 2 and284 AB
roots 4for accounts which ,1 is so then , ofroot double a is If
xP
However P(x) is a quartic which has a maximum of 4 roots
Thus no roots can have a multiplicity > 2
ABxAxxxP 234 23
and 1 , be roots let the
A431
c) Deduce that if is a multiple root, then its multiplicity is 2 and284 AB
roots 4for accounts which ,1 is so then , ofroot double a is If
xP
However P(x) is a quartic which has a maximum of 4 roots
Thus no roots can have a multiplicity > 2
ABxAxxxP 234 23
and 1 , be roots let the
A431
)1(rootsofsum
c) Deduce that if is a multiple root, then its multiplicity is 2 and284 AB
roots 4for accounts which ,1 is so then , ofroot double a is If
xP
However P(x) is a quartic which has a maximum of 4 roots
Thus no roots can have a multiplicity > 2
ABxAxxxP 234 23
and 1 , be roots let the
A431
)1(rootsofsum
B211
c) Deduce that if is a multiple root, then its multiplicity is 2 and284 AB
roots 4for accounts which ,1 is so then , ofroot double a is If
xP
However P(x) is a quartic which has a maximum of 4 roots
Thus no roots can have a multiplicity > 2
ABxAxxxP 234 23
and 1 , be roots let the
A431
)1(rootsofsum
B211
)2(
c) Deduce that if is a multiple root, then its multiplicity is 2 and284 AB
roots 4for accounts which ,1 is so then , ofroot double a is If
xP
However P(x) is a quartic which has a maximum of 4 roots
Thus no roots can have a multiplicity > 2
ABxAxxxP 234 23
and 1 , be roots let the
A431
)1(rootsofsum
B211
)2(
A41
c) Deduce that if is a multiple root, then its multiplicity is 2 and284 AB
roots 4for accounts which ,1 is so then , ofroot double a is If
xP
However P(x) is a quartic which has a maximum of 4 roots
Thus no roots can have a multiplicity > 2
ABxAxxxP 234 23
and 1 , be roots let the
A431
)1(rootsofsum
B211
)2(
A41
)3(
Substitute (3) into (1)
Substitute (3) into (1)
A
AA
211
43
411
Substitute (3) into (1)
A
AA
211
43
411
Substitute (3) into (2)
Substitute (3) into (1)
A
AA
211
43
411
Substitute (3) into (2)
BA
BA
BA
BAA
48 21
811
211
41-1
211
41
411
2
2
Substitute (3) into (1)
A
AA
211
43
411
Substitute (3) into (2)
BA
BA
BA
BAA
48 21
811
211
41-1
211
41
411
2
2
Exercise 5A; evens
Exercise 5B; 2, 4, 5b, 6b, 7b, 8 a,c,e,g,h
Substitute (3) into (1)
A
AA
211
43
411
Substitute (3) into (2)
BA
BA
BA
BAA
48 21
811
211
41-1
211
41
411
2
2
Exercise 5A; evens
Exercise 5B; 2, 4, 5b, 6b, 7b, 8 a,c,e,g,h
Note: tangent to a cubic has two solutions only.
Substitute (3) into (1)
A
AA
211
43
411
Substitute (3) into (2)
BA
BA
BA
BAA
48 21
811
211
41-1
211
41
411
2
2
Exercise 5A; evens
Exercise 5B; 2, 4, 5b, 6b, 7b, 8 a,c,e,g,h
Note: tangent to a cubic has two solutions only.
Substitute (3) into (1)
A
AA
211
43
411
Substitute (3) into (2)
BA
BA
BA
BAA
48 21
811
211
41-1
211
41
411
2
2
Exercise 5A; evens
Exercise 5B; 2, 4, 5b, 6b, 7b, 8 a,c,e,g,h
Note: tangent to a cubic has two solutions only. A double root
Substitute (3) into (1)
A
AA
211
43
411
Substitute (3) into (2)
BA
BA
BA
BAA
48 21
811
211
41-1
211
41
411
2
2
Exercise 5A; evens
Exercise 5B; 2, 4, 5b, 6b, 7b, 8 a,c,e,g,h
Note: tangent to a cubic has two solutions only. A double root
and a single root