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# X2 t02 04 forming polynomials (2013)

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### X2 t02 04 forming polynomials (2013)

1. 1. Forming Polynomials With The Roots Of Another
2. 2. Forming Polynomials With The Roots Of Another If  ,  ,  ,  are the roots of a polynomial, to form an equation with roots;
3. 3. Forming Polynomials With The Roots Of Another If  ,  ,  ,  are the roots of a polynomial, to form an equation with roots; 1 1 1 (1) , , ,   
4. 4. Forming Polynomials With The Roots Of Another If  ,  ,  ,  are the roots of a polynomial, to form an equation with roots; 1 1 1 (1) , , ,    let y  1 1 and substitute x  x y
5. 5. Forming Polynomials With The Roots Of Another If  ,  ,  ,  are the roots of a polynomial, to form an equation with roots; 1 1 1 (1) , , ,    (2) k , k , k , let y  1 1 and substitute x  x y
6. 6. Forming Polynomials With The Roots Of Another If  ,  ,  ,  are the roots of a polynomial, to form an equation with roots; 1 1 1 (1) , , ,    (2) k , k , k , let y  1 1 and substitute x  x y y let y  kx and substitute x  k
7. 7. Forming Polynomials With The Roots Of Another If  ,  ,  ,  are the roots of a polynomial, to form an equation with roots; 1 1 1 (1) , , ,    (2) k , k , k , (3)   c,   c,   c,  let y  1 1 and substitute x  x y y let y  kx and substitute x  k
8. 8. Forming Polynomials With The Roots Of Another If  ,  ,  ,  are the roots of a polynomial, to form an equation with roots; 1 1 1 (1) , , ,    let y  1 1 and substitute x  x y (2) k , k , k , y let y  kx and substitute x  k (3)   c,   c,   c,  let y  x  c and substitute x  y  c
9. 9. Forming Polynomials With The Roots Of Another If  ,  ,  ,  are the roots of a polynomial, to form an equation with roots; 1 1 1 (1) , , ,    let y  1 1 and substitute x  x y (2) k , k , k , y let y  kx and substitute x  k (3)   c,   c,   c,  let y  x  c and substitute x  y  c ( 4)  2 ,  2 ,  2 , 
10. 10. Forming Polynomials With The Roots Of Another If  ,  ,  ,  are the roots of a polynomial, to form an equation with roots; 1 1 1 (1) , , ,    let y  1 1 and substitute x  x y (2) k , k , k , y let y  kx and substitute x  k (3)   c,   c,   c,  let y  x  c and substitute x  y  c ( 4)  ,  ,  ,  2 2 2 let y  x 2 and substitute x  y 1 2
11. 11. e.g. If  ,  ,  are the roots of x 3  x  2  0, form an equation whose roots are;
12. 12. e.g. If  ,  ,  are the roots of x 3  x  2  0, form an equation whose roots are; a) 1 1 1 , ,   
13. 13. e.g. If  ,  ,  are the roots of x 3  x  2  0, form an equation whose roots are; a) 1 1 1 , ,    1 x 1 x y let y 
14. 14. e.g. If  ,  ,  are the roots of x 3  x  2  0, form an equation whose roots are; a) 1 1 1 , ,    1 x 1 x y let y  3 1 1    20  y y
15. 15. e.g. If  ,  ,  are the roots of x 3  x  2  0, form an equation whose roots are; a) 1 1 1 , ,    1 x 1 x y let y  3 1 1    20  y y 1  y 2  2 y3  0
16. 16. b)   1,   1,   1
17. 17. b)   1,   1,   1 let y  x  1 x  y 1
18. 18. b)   1,   1,   1 let y  x  1 x  y 1  y  13   y  1  2  0
19. 19. b)   1,   1,   1 let y  x  1 x  y 1  y  13   y  1  2  0 y3  3 y 2  3 y 1  y 1  2  0 y3  3 y2  4 y  0
20. 20. b)   1,   1,   1 let y  x  1 x  y 1  y  13   y  1  2  0 y3  3 y 2  3 y 1  y 1  2  0 y3  3 y2  4 y  0 c)  2 ,  2 ,  2
21. 21. b)   1,   1,   1 let y  x  1 x  y 1  y  13   y  1  2  0 y3  3 y 2  3 y 1  y 1  2  0 y3  3 y2  4 y  0 c)  2 ,  2 ,  2 let y  x 2 x y 1 2
22. 22. b)   1,   1,   1 let y  x  1 x  y 1  y  13   y  1  2  0 y3  3 y 2  3 y 1  y 1  2  0 y3  3 y2  4 y  0 c)  2 ,  2 ,  2 let y  x 2 x y 1 2  y   1 2 3 1    y2  2  0  
23. 23. b)   1,   1,   1 let y  x  1 x  y 1  y  13   y  1  2  0 y3  3 y 2  3 y 1  y 1  2  0 y3  3 y2  4 y  0 c)  2 ,  2 ,  2 let y  x 2 x y 1 2  y   1 2 3 1    y2  2  0   3 2 1 2 y  y 20
24. 24. b)   1,   1,   1 let y  x  1 x  y 1  y  13   y  1  2  0 y3  3 y 2  3 y 1  y 1  2  0 y3  3 y2  4 y  0 c)  2 ,  2 ,  2 let y  x 2 x y 1 2  y   1 2 3 1    y2  2  0   3 2 1 2 y  y 20 1 2 y  y  1  2
25. 25. b)   1,   1,   1 let y  x  1 x  y 1  y  13   y  1  2  0 y3  3 y 2  3 y 1  y 1  2  0 y3  3 y2  4 y  0 c)  2 ,  2 ,  2 let y  x 2 x y 1 2  y   1 2 3 1    y2  2  0   3 2 1 2 y  y 20 1 2 y  y  1  2 y  y  1  4 2
26. 26. b)   1,   1,   1 let y  x  1 x  y 1  y  13   y  1  2  0 y3  3 y 2  3 y 1  y 1  2  0 y3  3 y2  4 y  0 c)  2 ,  2 ,  2 let y  x 2 x y 1 2  y   1 2 3 1    y2  2  0   3 2 1 2 y  y 20 1 2 y  y  1  2 y  y  1  4 2 y3  2 y 2  y  4 y3  2 y 2  y  4  0
27. 27. d) 1  2 , 1  2 , 1 2
28. 28. d) 1  2 , 1  2 let y  , 1 2 1 x2 x y  1 2
29. 29. d) 1  2 , 1  2 , 1 2 1 let y  2 x x y  y 1 2  3 2 y  1 2 20
30. 30. d) 1  2 , 1  2 , 1 2 1 let y  2 x x y 1  2 y  3 2 y y  3 2  1 2 20  y  1  2
31. 31. d) 1  2 , 1  2 , 1 2 1 let y  2 x x y 1  2 y  3 2 y y  3 2  1 2 20  y  1  2  y  1  2 y 3 2
32. 32. d) 1  2 , 1  2 , 1 2 1 let y  2 x x y 1  2 y  3 2 y y  3 2  1 2 20  y  1  2  y  1  2 y  y  12  4 y 3 3 2
33. 33. d) 1  2 , 1  2 , 1 2 1 let y  2 x x y 1  2 y  3 2 y y  3 2  1 2 20  y  1  2  y  1  2 y  y  12  4 y 3 y 2  2 y 1  4 y3 4 y3  y2  2 y 1  0 3 2
34. 34. e) Find  2   2   2
35. 35. e) Find  2   2   2 2   2  2      2  2
36. 36. e) Find  2   2   2 2   2  2      2  2  0   21  2 2
37. 37. e) Find  2   2   2 2   2  2      2  2  0   21  2 2 OR using equation found in c)
38. 38. e) Find  2   2   2 2   2  2      2  2  0   21  2 2 OR using equation found in c) 2   2  2 b  a
39. 39. e) Find  2   2   2 2   2  2      2  2  0   21  2 2 OR using equation found in c) 2   2  2 b  a 2  1  2
40. 40. e) Find  2   2   2 2   2  2      2  2  0   21  2 2 OR using equation found in c) 2   2  2 b  a 2  1  2 Cambridge: Exercise 5C; 1 to 11, 13, 14, 15 Patel: Exercise 5E; 9, 10, 11, 14, 16, 18, 19, 23, 24, 27, 30, 32, 34, 35