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X2 t02 02 multiple roots (2013)

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X2 t02 02 multiple roots (2013)

1. 1. Multiple Roots
2. 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. 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. 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. 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  m1   x  a  R x  (where x = a is not a root of R(x)) m m 1
6. 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  m1   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. 7. e.g. (i) Solve the equation x 3  4 x 2  3 x  18  0 , given that it has a double root
8. 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. 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. 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. 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  32  x  2  0
12. 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  32  x  2  0 x  2 or x  3
13. 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. 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 P0   1  0,   0
15. 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 P0   1  0,   0 P1  1  A  B  A  1  2A  B  2
16. 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 P0   1  0,   0 P1  1  A  B  A  1  2A  B  2 P 1  1  A  B  A  1  2 A  B  2
17. 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 P0   1  0,   0 P1  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. 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 P0   1  0,   0 P1  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. 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 P0   1  0,   0 P1  1  A  B  A  1  2A  B  2 BUT 2  B 2  4 A2 2  B  2 A  2A  B  2  0  P1  0, P 1  0 hence   1 P 1  1  A  B  A  1  2 A  B  2
20. 20. b) Show that 1  is a root
21. 21. b) Show that 1  is a root  1   1  A  B  A 1 P  4 3 2      
22. 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. 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. 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. 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. 26. c) Deduce that if  is a multiple root, then its multiplicity is 2 and 4 B  8  A2
27. 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. 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. 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. 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. 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 
32. 32. Substitute (3) into (1)
33. 33. Substitute (3) into (1)  1 1 3  A A  4 4 1 1   A  2
34. 34. Substitute (3) into (1)  1 1 3  A A  4 4 1 1   A  2 Substitute (3) into (2)
35. 35. Substitute (3) into (1)  1 1 3  A A  4 4 1 1   A  2 Substitute (3) into (2) 1 1 1 1 1  A  A  B 4 4  2 1 1 1 1 - A    B   4   2 1 1 1  A2  B 8 2 8  A2  4 B
36. 36. Substitute (3) into (1)  1 1 3  A A  4 4 1 1   A  2 Substitute (3) into (2) 1 1 1 1 1  A  A  B 4 4  2 1 1 1 1 - A    B   4   2 1 1 1  A2  B 8 2 8  A2  4 B Cambridge: Exercise 5B; 1 to 16 Patel: Exercise 5B; 6b, 7b, 8 a,c,e,g,h
37. 37. Substitute (3) into (1)  1 1 3  A A  4 4 1 1   A  2 Substitute (3) into (2) 1 1 1 1 1  A  A  B 4 4  2 1 1 1 1 - A    B   4   2 1 1 1  A2  B 8 2 8  A2  4 B Cambridge: Exercise 5B; 1 to 16 Patel: Exercise 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