1. Forming Polynomials With
The Roots Of Another

2. Forming Polynomials With
The Roots Of Another
If  ,  ,  ,  are the roots of a polynomial, to form an equation
with roots;

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. 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. 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. 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. 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. 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. 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. 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. e.g. If  ,  ,  are the roots of x 3  x  2  0, form an equation
whose roots are;

12. e.g. If  ,  ,  are the roots of x 3  x  2  0, form an equation
whose roots are;
a)
1 1 1
, ,
  

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. 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. 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. b)   1,   1,   1

17. b)   1,   1,   1
let y  x  1
x  y 1

18. b)   1,   1,   1
let y  x  1
x  y 1
 y  13   y  1  2  0

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. 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. 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. 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. 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. 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. 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. 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. d)
1

2
,
1

2
,
1
2

28. d)
1

2
,
1

2
let y 
,
1
2
1
x2
x y

1
2

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. 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. 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. 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. 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. e) Find  2   2   2

35. e) Find  2   2   2
2   2  2
     2 
2

36. e) Find  2   2   2
2   2  2
     2 
2
 0   21
 2
2

37. e) Find  2   2   2
2   2  2
     2 
2
 0   21
 2
2
OR using equation found in c)

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. 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. 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