1. Roots and Coefficients

2. Roots and Coefficients
Quadratics
ax 2  bx  c  0

3. Roots and Coefficients
Quadratics
ax 2  bx  c  0
b
  
a

4. Roots and Coefficients
Quadratics
ax 2  bx  c  0
b
  
a
c
 
a

5. Roots and Coefficients
Quadratics
ax 2  bx  c  0
b
  
a
Cubics
ax 3  bx 2  cx  d  0
c
 
a

6. Roots and Coefficients
Quadratics
ax 2  bx  c  0
b
  
a
Cubics
ax 3  bx 2  cx  d  0
     
b
a
c
 
a

7. Roots and Coefficients
Quadratics
ax 2  bx  c  0
c
 
a
b
  
a
Cubics
ax 3  bx 2  cx  d  0
b
     
a
c
     
a

8. Roots and Coefficients
Quadratics
ax 2  bx  c  0
c
 
a
b
  
a
Cubics
ax 3  bx 2  cx  d  0
b
     
a
d
  
a
c
     
a

9. Roots and Coefficients
Quadratics
ax 2  bx  c  0
c
 
a
b
  
a
Cubics
ax 3  bx 2  cx  d  0
b
     
a
Quartics
d
  
a
c
     
a
ax 4  bx 3  cx 2  dx  e  0

10. Roots and Coefficients
Quadratics
ax 2  bx  c  0
c
 
a
b
  
a
Cubics
ax 3  bx 2  cx  d  0
b
     
a
Quartics
d
  
a
c
     
a
ax 4  bx 3  cx 2  dx  e  0
b
      
a

11. Roots and Coefficients
Quadratics
ax 2  bx  c  0
c
 
a
b
  
a
Cubics
ax 3  bx 2  cx  d  0
b
     
a
Quartics
d
  
a
c
     
a
ax 4  bx 3  cx 2  dx  e  0
b
      
a
c
           
a

12. Roots and Coefficients
Quadratics
ax 2  bx  c  0
c
 
a
b
  
a
Cubics
ax 3  bx 2  cx  d  0
b
     
a
Quartics
d
  
a
c
     
a
ax 4  bx 3  cx 2  dx  e  0
c
b
           
      
a
a
d
        
a

13. Roots and Coefficients
Quadratics
ax 2  bx  c  0
c
 
a
b
  
a
Cubics
ax 3  bx 2  cx  d  0
b
     
a
Quartics
d
  
a
c
     
a
ax 4  bx 3  cx 2  dx  e  0
c
b
           
      
a
a
d
e
        
 
a
a

14. For the polynomial equation;
ax n  bx n1  cx n2  dx n3    0

15. For the polynomial equation;
ax n  bx n1  cx n2  dx n3    0
b
   a
(sum of roots, one at a time)

16. For the polynomial equation;
ax n  bx n1  cx n2  dx n3    0
b
   a
c
  a
(sum of roots, one at a time)
(sum of roots, two at a time)

17. For the polynomial equation;
ax n  bx n1  cx n2  dx n3    0
b
   a
c
  a
d
   a
(sum of roots, one at a time)
(sum of roots, two at a time)
(sum of roots, three at a time)

18. For the polynomial equation;
ax n  bx n1  cx n2  dx n3    0




b

a
c

a
d

a
e

a
(sum of roots, one at a time)
(sum of roots, two at a time)
(sum of roots, three at a time)
(sum of roots, four at a time)

19. For the polynomial equation;
ax n  bx n1  cx n2  dx n3    0




b

a
c

a
d

a
e

a
Note:

(sum of roots, one at a time)
(sum of roots, two at a time)
(sum of roots, three at a time)
(sum of roots, four at a time)
2
     2 
2

20. e.g. (i ) If  ,  and  are the roots of 2 x 3  5 x 2  3 x  1  0, find the
values of;
a) 4  4   4  7

21. e.g. (i ) If  ,  and  are the roots of 2 x 3  5 x 2  3 x  1  0, find the
values of;
a) 4  4   4  7
5
    
2

22. e.g. (i ) If  ,  and  are the roots of 2 x 3  5 x 2  3 x  1  0, find the
values of;
a) 4  4   4  7
5
    
2
      
3
2

23. e.g. (i ) If  ,  and  are the roots of 2 x 3  5 x 2  3 x  1  0, find the
values of;
a) 4  4   4  7
5
    
2
      
3
2
  
1
2

24. e.g. (i ) If  ,  and  are the roots of 2 x 3  5 x 2  3 x  1  0, find the
values of;
a) 4  4   4  7
5
    
2
      
5  1
4  4   4  7  4    7   
2  2
3
2
  
1
2

25. e.g. (i ) If  ,  and  are the roots of 2 x 3  5 x 2  3 x  1  0, find the
values of;
a) 4  4   4  7
5
    
2
      
5  1
4  4   4  7  4    7   
2  2
27

2
3
2
  
1
2

26. e.g. (i ) If  ,  and  are the roots of 2 x 3  5 x 2  3 x  1  0, find the
values of;
a) 4  4   4  7
5
    
2
      
5  1
4  4   4  7  4    7   
2  2
27

2
1 1 1
b)  



3
2
  
1
2

27. e.g. (i ) If  ,  and  are the roots of 2 x 3  5 x 2  3 x  1  0, find the
values of;
a) 4  4   4  7
5
    
2
      
5  1
4  4   4  7  4    7   
2  2
27

2
1 1 1
    
b)  





3
2
  
1
2

28. e.g. (i ) If  ,  and  are the roots of 2 x 3  5 x 2  3 x  1  0, find the
values of;
a) 4  4   4  7
5
    
2
      
5  1
4  4   4  7  4    7   
2  2
27

2
1 1 1
    
b)  





3
 2
1

2

3
2
  
1
2

29. e.g. (i ) If  ,  and  are the roots of 2 x 3  5 x 2  3 x  1  0, find the
values of;
a) 4  4   4  7
5
    
2
      
5  1
4  4   4  7  4    7   
2  2
27

2
1 1 1
    
b)  





3
 2
1

2
 3

3
2
  
1
2

30. e.g. (i ) If  ,  and  are the roots of 2 x 3  5 x 2  3 x  1  0, find the
values of;
a) 4  4   4  7
5
    
2
      
3
2
  
5  1
4  4   4  7  4    7   
2  2
27

2
1 1 1
    
b)  

c)  2   2   2




3
 2
1

2
 3

1
2

31. e.g. (i ) If  ,  and  are the roots of 2 x 3  5 x 2  3 x  1  0, find the
values of;
a) 4  4   4  7
5
    
2
      
3
2
  
1
2
5  1
4  4   4  7  4    7   
2  2
27

2
1 1 1
    
b)  

c)  2   2   2
  
2

        2      
3

 2
1

2
 3

32. e.g. (i ) If  ,  and  are the roots of 2 x 3  5 x 2  3 x  1  0, find the
values of;
a) 4  4   4  7
5
    
2
      
3
2
  
1
2
5  1
4  4   4  7  4    7   
2  2
27

2
1 1 1
    
b)  

c)  2   2   2
  
2

        2      
3
2

5

 3
    2  
 2
2
 2
1

2
 3

33. e.g. (i ) If  ,  and  are the roots of 2 x 3  5 x 2  3 x  1  0, find the
values of;
a) 4  4   4  7
5
    
2
      
3
2
  
1
2
5  1
4  4   4  7  4    7   
2  2
27

2
1 1 1
    
b)  

c)  2   2   2
  
2

        2      
3
2

5

 3
    2  
 2
2
 2
1

37
2

4
 3

34. 1988 Extension 1 HSC Q2c)
3
If  ,  and  are the roots of x  3 x  1  0 find:
(i)     

35. 1988 Extension 1 HSC Q2c)
3
If  ,  and  are the roots of x  3 x  1  0 find:
(i)     
     0

36. 1988 Extension 1 HSC Q2c)
3
If  ,  and  are the roots of x  3 x  1  0 find:
(i)     
     0
(ii) 

37. 1988 Extension 1 HSC Q2c)
3
If  ,  and  are the roots of x  3 x  1  0 find:
(i)     
     0
(ii) 
  1

38. 1988 Extension 1 HSC Q2c)
3
If  ,  and  are the roots of x  3 x  1  0 find:
(i)     
     0
(ii) 
  1
(iii)
1


1


1


39. 1988 Extension 1 HSC Q2c)
3
If  ,  and  are the roots of x  3 x  1  0 find:
(i)     
     0
(ii) 
  1
(iii)
1


1


1

    
  
  

1
1
1

40. 1988 Extension 1 HSC Q2c)
3
If  ,  and  are the roots of x  3 x  1  0 find:
(i)     
     0
(ii) 
  1
(iii)
1


1


1

    
  
  

1
1
1

3
1

41. 1988 Extension 1 HSC Q2c)
3
If  ,  and  are the roots of x  3 x  1  0 find:
(i)     
     0
(ii) 
  1
(iii)
1


1


1

    
  
  

1
1
1
3
1
3


42. 2003 Extension 1 HSC Q4c)
It is known that two of the roots of the equation 2 x 3  x 2  kx  6  0 are
reciprocals of each other.
Find the value of k.

43. 2003 Extension 1 HSC Q4c)
It is known that two of the roots of the equation 2 x 3  x 2  kx  6  0 are
reciprocals of each other.
Find the value of k.
1
Let the roots be  , and 


44. 2003 Extension 1 HSC Q4c)
It is known that two of the roots of the equation 2 x 3  x 2  kx  6  0 are
reciprocals of each other.
Find the value of k.
1
Let the roots be  , and 
 1      6
  
2
 


45. 2003 Extension 1 HSC Q4c)
It is known that two of the roots of the equation 2 x 3  x 2  kx  6  0 are
reciprocals of each other.
Find the value of k.
1
Let the roots be  , and 
 1      6
  
2
 
  3


46. 2003 Extension 1 HSC Q4c)
It is known that two of the roots of the equation 2 x 3  x 2  kx  6  0 are
reciprocals of each other.
Find the value of k.
1
Let the roots be  , and 
 1      6
  
2
 
  3

P   3  0

47. 2003 Extension 1 HSC Q4c)
It is known that two of the roots of the equation 2 x 3  x 2  kx  6  0 are
reciprocals of each other.
Find the value of k.
1
Let the roots be  , and 
 1      6
  
2
 
  3

P   3  0
2 3   3  k  3  6  0
3
2

48. 2003 Extension 1 HSC Q4c)
It is known that two of the roots of the equation 2 x 3  x 2  kx  6  0 are
reciprocals of each other.
Find the value of k.
1
Let the roots be  , and 
 1      6
  
2
 
  3

P   3  0
2 3   3  k  3  6  0
3
2
 54  9  3k  6  0

49. 2003 Extension 1 HSC Q4c)
It is known that two of the roots of the equation 2 x 3  x 2  kx  6  0 are
reciprocals of each other.
Find the value of k.
1
Let the roots be  , and 
 1      6
  
2
 
  3

P   3  0
2 3   3  k  3  6  0
3
2
 54  9  3k  6  0
3k  39

50. 2003 Extension 1 HSC Q4c)
It is known that two of the roots of the equation 2 x 3  x 2  kx  6  0 are
reciprocals of each other.
Find the value of k.
1
Let the roots be  , and 
 1      6
  
2
 
  3

P   3  0
2 3   3  k  3  6  0
3
2
 54  9  3k  6  0
3k  39
k  13

51. 2006 Extension 1 HSC Q4a)
The cubic polynomial P x   x 3  rx 2  sx  t , where r, s and t are real
numbers, has three real zeros, 1,  and  
(i) Find the value of r

52. 2006 Extension 1 HSC Q4a)
The cubic polynomial P x   x 3  rx 2  sx  t , where r, s and t are real
numbers, has three real zeros, 1,  and  
(i) Find the value of r
1       r

53. 2006 Extension 1 HSC Q4a)
The cubic polynomial P x   x 3  rx 2  sx  t , where r, s and t are real
numbers, has three real zeros, 1,  and  
(i) Find the value of r
1       r
r  1

54. 2006 Extension 1 HSC Q4a)
The cubic polynomial P x   x 3  rx 2  sx  t , where r, s and t are real
numbers, has three real zeros, 1,  and  
(i) Find the value of r
1       r
r  1
(ii) Find the value of s + t

55. 2006 Extension 1 HSC Q4a)
The cubic polynomial P x   x 3  rx 2  sx  t , where r, s and t are real
numbers, has three real zeros, 1,  and  
(i) Find the value of r
1       r
r  1
(ii) Find the value of s + t
1   1         s

56. 2006 Extension 1 HSC Q4a)
The cubic polynomial P x   x 3  rx 2  sx  t , where r, s and t are real
numbers, has three real zeros, 1,  and  
(i) Find the value of r
1       r
r  1
(ii) Find the value of s + t
1   1         s
s   2

57. 2006 Extension 1 HSC Q4a)
The cubic polynomial P x   x 3  rx 2  sx  t , where r, s and t are real
numbers, has three real zeros, 1,  and  
(i) Find the value of r
1       r
r  1
(ii) Find the value of s + t
1   1         s
s   2
1     t

58. 2006 Extension 1 HSC Q4a)
The cubic polynomial P x   x 3  rx 2  sx  t , where r, s and t are real
numbers, has three real zeros, 1,  and  
(i) Find the value of r
1       r
r  1
(ii) Find the value of s + t
1   1         s
s   2
1     t
t 2

59. 2006 Extension 1 HSC Q4a)
The cubic polynomial P x   x 3  rx 2  sx  t , where r, s and t are real
numbers, has three real zeros, 1,  and  
(i) Find the value of r
1       r
r  1
(ii) Find the value of s + t
1   1         s
s   2
1     t
t 2
s  t  0

60. Exercise 4F; 2, 4, 5ac, 6ac, 8, 10a, 13, 15,
16ad, 17, 18, 19