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 1
(1) , , ,  let y  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 1
(1) , , ,  let y  and substitute x 
   x y
(2) k , k , k ,

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 1
(1) , , ,  let y  and substitute x 
   x y
y
(2) k , k , k , 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 1
(1) , , ,  let y  and substitute x 
   x y
y
(2) k , k , k , let y  kx and substitute x 
k
(3)   c,   c,   c, 

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 1
(1) , , ,  let y  and substitute x 
   x y
y
(2) k , k , k , 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 1
(1) , , ,  let y  and substitute x 
   x y
y
(2) k , k , k , 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 1
(1) , , ,  let y  and substitute x 
   x y
y
(2) k , k , k , let y  kx and substitute x 
k
(3)   c,   c,   c,  let y  x  c and substitute x  y  c
1
( 4)  ,  ,  , 
2 2 2
let y  x 2 and substitute x  y 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;
1 1 1
a) , ,
  

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

14. e.g. If  ,  ,  are the roots of x 3  x  2  0, form an equation
whose roots are;
1 1 1
a) , ,
  
3
1 1 1
let y     20
x  y y
1
x
y

15. e.g. If  ,  ,  are the roots of x 3  x  2  0, form an equation
whose roots are;
1 1 1
a) , ,
  
3
1 1 1
let y     20
x  y y
1
x 1  y2  2 y3  0
y

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  y  13   y  1  2  0
x  y 1

19. b)   1,   1,   1
let y  x  1  y  13   y  1  2  0
x  y 1
y3  3 y 2  3 y 1  y 1  2  0
y3  3 y 2  4 y  0

20. b)   1,   1,   1
let y  x  1  y  13   y  1  2  0
x  y 1
y3  3 y 2  3 y 1  y 1  2  0
y3  3 y 2  4 y  0
c)  2 ,  2 ,  2

21. b)   1,   1,   1
let y  x  1  y  13   y  1  2  0
x  y 1
y3  3 y 2  3 y 1  y 1  2  0
y3  3 y 2  4 y  0
c)  2 ,  2 ,  2
let y  x 2
1
x y 2

22. b)   1,   1,   1
let y  x  1  y  13   y  1  2  0
x  y 1
y3  3 y 2  3 y 1  y 1  2  0
y3  3 y2  4 y  0
c)  2 ,  2 ,  2  1

3 1
let y  x 2 y

2   y2  2  0

1
 
x y 2

23. b)   1,   1,   1
let y  x  1  y  13   y  1  2  0
x  y 1
y3  3 y 2  3 y 1  y 1  2  0
y3  3 y2  4 y  0
c)  2 ,  2 ,  2  1

3 1
let y  x 2 y

2   y2  2  0

1
 
3 1
x y 2
y  y 20
2 2

24. b)   1,   1,   1
let y  x  1  y  13   y  1  2  0
x  y 1
y3  3 y 2  3 y 1  y 1  2  0
y3  3 y2  4 y  0
c)  2 ,  2 ,  2  1

3 1
let y  x 2 y

2   y2  2  0

1
 
3 1
x y 2
y  y 20
2 2
1
y  y  1  2
2

25. b)   1,   1,   1
let y  x  1  y  13   y  1  2  0
x  y 1
y3  3 y 2  3 y 1  y 1  2  0
y3  3 y2  4 y  0
c)  2 ,  2 ,  2  1

3 1
let y  x 2 y

2   y2  2  0

1
 
3 1
x y 2
y  y 20
2 2
1
y  y  1  2
2
y  y  1  4
2

26. b)   1,   1,   1
let y  x  1  y  13   y  1  2  0
x  y 1
y3  3 y 2  3 y 1  y 1  2  0
y3  3 y2  4 y  0
c)  2 ,  2 ,  2  1

3 1
let y  x 2 y

2   y2  2  0

1
 
3 1
x y 2
y  y 20
2 2
1
y  y  1  2
2
y  y  1  4
2
y3  2 y 2  y  4
y3  2 y 2  y  4  0

27. 1 1 1
d) , ,
 2
 2
2

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

29. 1 1 1
d) , ,
 2
 2
2
1 3 1
let y  2 y

2
y

2
20
x
1

x y 2

30. 1 1 1
d) , ,
 2
 2
2
1 3 1
let y  2 y

2
y

2
20
x
3

 y  1  2
1
 2
x y 2 y

31. 1 1 1
d) , ,
 2
 2
2
1 3 1
let y  2 y

2
y

2
20
x
3

 y  1  2
1
 2
x y 2 y
3
 y  1  2 y 2

32. 1 1 1
d) , ,
 2
 2
2
1 3 1
let y  2 y

2
y

2
20
x
3

 y  1  2
1
 2
x y 2 y
3
 y  1  2 y 2
 y  12  4 y 3

33. 1 1 1
d) , ,
 2
 2
2
1 3 1
let y  2 y

2
y

2
20
x
3

 y  1  2
1
 2
x y 2 y
3
 y  1  2 y 2
 y  12  4 y 3
y2  2 y 1  4 y3
4 y3  y2  2 y 1  0

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 OR using equation found in c)
     2 
2
 0   21
2
 2

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

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

40. e) Find  2   2   2
2   2  2 OR using equation found in c)
     2  2   2  2
2
b
 0   21
2 
a
 2 2

1
 2
Exercise 5D; 10 to 16
Exercise 5E; 2cd, 4b, 7, 9, 10, 18, 27, 30, 34, 35