The document discusses the relationships between the roots and coefficients of polynomial equations. It provides formulas to find the sum of roots taken one, two, three, or more at a time in terms of the coefficients. For a polynomial of degree n in the form ax^n + bx^(n-1) + cx^(n-2) + ..., the formulas are provided to find the sum of roots one at a time as -b/a, two at a time as c/a, three at a time as -d/a, and so on. An example is also given to demonstrate using the formulas.

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