The document discusses the relationships between the roots and coefficients of polynomial equations. It provides formulas to relate the sum of the roots taken one, two, three, or more at a time to the coefficients of the polynomial. As the degree of the polynomial increases, more terms are needed in the formulas. An example problem demonstrates how to use the formulas to calculate values involving the roots of a cubic polynomial.

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 c
    
a a

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

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

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

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

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

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

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

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

13. Roots and Coefficients
Quadratics ax 2  bx  c  0
b c
    
a a
Cubics ax 3  bx 2  cx  d  0
b c
           
a a
d
  
a
Quartics ax 4  bx 3  cx 2  dx  e  0
b c
                  
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 (sum of roots, one at a time)
c
  a (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 (sum of roots, one at a time)
c
  a (sum of roots, two at a time)
d
   a (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
(sum of roots, one at a time)
c
 
a
(sum of roots, two at a time)
d
 
a
(sum of roots, three at a time)
e
 
a
(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
(sum of roots, one at a time)
c
 
a
(sum of roots, two at a time)
d
 
a
(sum of roots, three at a time)
e
 
a
(sum of roots, four at a time)
Note:
      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 3
           
2 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 3 1
              
2 2 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 3 1
              
2 2 2
5  1
4  4   4  7  4    7   
2  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 3 1
              
2 2 2
5  1
4  4   4  7  4    7   
2  2
27

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 3 1
              
2 2 2
5  1
4  4   4  7  4    7   
2  2
27

2
1 1 1
b)  
  

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 3 1
              
2 2 2
5  1
4  4   4  7  4    7   
2  2
27

2
1 1 1     
b)   
   

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 3 1
              
2 2 2
5  1
4  4   4  7  4    7   
2  2
27

2
1 1 1     
b)   
   
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 3 1
              
2 2 2
5  1
4  4   4  7  4    7   
2  2
27

2
1 1 1     
b)   
   
3

 2
1

2
 3

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 3 1
              
2 2 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

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 3 1
              
2 2 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 3 1
              
2 2 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  
1 2  2

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 3 1
              
2 2 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  
1 2  2
 37
2 
 3 4

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

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

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

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

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

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

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

41. 1988 Extension 1 HSC Q2c)
If  ,  and  are the roots of x  3 x  1  0 find:
3
(i)     
     0
(ii) 
  1
1 1 1
(iii)  
  
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
   P   3  0
  2
  3

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
   P   3  0
  2
  3 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
   P   3  0
  2
  3 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
   P   3  0
  2
  3 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
   P   3  0
  2
  3 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 1     t
s   2

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 1     t
s   2 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 1     t
s   2 t 2
s  t  0

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