x2 t02 01 multiple roots (2011)
TRANSCRIPT
![Page 1: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/1.jpg)
Polynomials
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PolynomialsIf (x – a) is a factor of P(x), then P(a) = 0
![Page 3: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/3.jpg)
PolynomialsIf (x – a) is a factor of P(x), then P(a) = 0
If (ax – b) is a factor of P(x), then = 0
![Page 4: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/4.jpg)
PolynomialsIf (x – a) is a factor of P(x), then P(a) = 0
abPIf (ax – b) is a factor of P(x), then = 0
![Page 5: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/5.jpg)
Multiple Roots
![Page 6: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/6.jpg)
Multiple RootsIf P(x), has a root, x = a, of multiplicity m,
then P’(x) has a root, x = a, of multiplicity m - 1
![Page 7: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/7.jpg)
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
![Page 8: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/8.jpg)
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))
![Page 9: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/9.jpg)
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))
![Page 10: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/10.jpg)
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))
![Page 11: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/11.jpg)
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))
![Page 12: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/12.jpg)
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))
![Page 13: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/13.jpg)
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
![Page 14: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/14.jpg)
e.g. (i) Solve the equation , given that it has a double root
01834 23 xxx
![Page 15: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/15.jpg)
e.g. (i) Solve the equation , given that it has a double root
01834 23 xxx
1834 23 xxxxP
![Page 16: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/16.jpg)
e.g. (i) Solve the equation , given that it has a double root
01834 23 xxx
1834 23 xxxxP 383 2 xxxP
![Page 17: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/17.jpg)
e.g. (i) Solve the equation , given that it has a double root
01834 23 xxx
1834 23 xxxxP 383 2 xxxP
313 xx
![Page 18: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/18.jpg)
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
![Page 19: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/19.jpg)
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
![Page 20: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/20.jpg)
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
![Page 21: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/21.jpg)
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
![Page 22: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/22.jpg)
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
![Page 23: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/23.jpg)
(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
![Page 24: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/24.jpg)
(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
![Page 25: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/25.jpg)
(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
![Page 26: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/26.jpg)
(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
![Page 27: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/27.jpg)
(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
![Page 28: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/28.jpg)
(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
![Page 29: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/29.jpg)
(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
![Page 30: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/30.jpg)
(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
![Page 31: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/31.jpg)
(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
![Page 32: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/32.jpg)
b) Show that is a root1
![Page 33: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/33.jpg)
b) Show that is a root1
111234
ABAP
![Page 34: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/34.jpg)
b) Show that is a root1
111234
ABAP
4
4321
ABA
![Page 35: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/35.jpg)
b) Show that is a root1
111234
ABAP
4
4321
ABA
4P
![Page 36: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/36.jpg)
b) Show that is a root1
111234
ABAP
4
4321
ABA
4P
40
![Page 37: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/37.jpg)
b) Show that is a root1
111234
ABAP
4
4321
ABA
4P
40
root)aisas 0( P
![Page 38: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/38.jpg)
b) Show that is a root1
111234
ABAP
4
4321
ABA
4P
40
root)aisas 0( P
0
![Page 39: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/39.jpg)
b) Show that is a root1
111234
ABAP
4
4321
ABA
4P
40
root)aisas 0( P
0
xP ofroot a is 1
![Page 40: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/40.jpg)
c) Deduce that if is a multiple root, then its multiplicity is 2 and284 AB
![Page 41: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/41.jpg)
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
![Page 42: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/42.jpg)
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
![Page 43: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/43.jpg)
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
![Page 44: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/44.jpg)
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
![Page 45: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/45.jpg)
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
![Page 46: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/46.jpg)
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
![Page 47: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/47.jpg)
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
![Page 48: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/48.jpg)
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
![Page 49: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/49.jpg)
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(
![Page 50: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/50.jpg)
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
![Page 51: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/51.jpg)
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(
![Page 52: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/52.jpg)
Substitute (3) into (1)
![Page 53: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/53.jpg)
Substitute (3) into (1)
A
AA
211
43
411
![Page 54: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/54.jpg)
Substitute (3) into (1)
A
AA
211
43
411
Substitute (3) into (2)
![Page 55: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/55.jpg)
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
![Page 56: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/56.jpg)
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
![Page 57: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/57.jpg)
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.
![Page 58: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/58.jpg)
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.
![Page 59: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/59.jpg)
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
![Page 60: X2 T02 01 multiple roots (2011)](https://reader030.vdocuments.net/reader030/viewer/2022032714/55ab00ef1a28ab8e288b476a/html5/thumbnails/60.jpg)
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