The document discusses finding the greatest term in polynomial expansions. It provides an example of finding the greatest coefficient in the expansion of (2 + 3x)^20. Through algebraic steps, it is shown that the greatest coefficient is T13 = 20C12 * 2831^2. It then gives an example of finding the greatest term in the expansion of (3x - 4)^15 when x = 1/2. After setting up expressions for the terms Tk+1 and Tk, it derives an inequality to determine the value of k that makes Tk+1 the greatest term.

1. Greatest Coefficients and
Greatest Terms

2. Greatest Coefficients and
Greatest Terms
If Tk 1  Tk then Tk 1 is the greatest term

3. Greatest Coefficients and
Greatest Terms
If Tk 1  Tk then Tk 1 is the greatest term
e.g . For the expansion 2  3 x  find the greatest coefficient
20

4. Greatest Coefficients and
Greatest Terms
If Tk 1  Tk then Tk 1 is the greatest term
e.g . For the expansion 2  3 x  find the greatest coefficient
20
Tk 1  20Ck 2  3 x 
20 k k

5. Greatest Coefficients and
Greatest Terms
If Tk 1  Tk then Tk 1 is the greatest term
e.g . For the expansion 2  3 x  find the greatest coefficient
20
Tk 1  20Ck 2  3 x 
20 k k
Tk  20Ck 1 2  3x k 1
21 k

6. Greatest Coefficients and
Greatest Terms
If Tk 1  Tk then Tk 1 is the greatest term
e.g . For the expansion 2  3 x  find the greatest coefficient
20
Tk 1  20Ck 2  3 x 
20 k k
Tk  20Ck 1 2  3x k 1
21 k
If Tk 1  Tk then Tk 1 is the greatest term

7. Greatest Coefficients and
Greatest Terms
If Tk 1  Tk then Tk 1 is the greatest term
e.g . For the expansion 2  3 x  find the greatest coefficient
20
Tk 1  20Ck 2  3 x 
20 k k
Tk  20Ck 1 2  3x k 1
21 k
If Tk 1  Tk then Tk 1 is the greatest term
Ck 2  3  20Ck 1 2  3
20 20 k k 21k k 1

8. Greatest Coefficients and
Greatest Terms
If Tk 1  Tk then Tk 1 is the greatest term
e.g . For the expansion 2  3 x  find the greatest coefficient
20
Tk 1  20Ck 2  3 x 
20 k k
Tk  20Ck 1 2  3x k 1
21 k
If Tk 1  Tk then Tk 1 is the greatest term
Ck 2  3  20Ck 1 2  3
20 20 k k 21k k 1
Ck 2  3
20 20k k
1
Ck 1 2  3
20 21k k 1

9. Ck 2  3
20 20k k
1
Ck 1 2  3
20 21k k 1

10. Ck 2  3
20 20k k
1
Ck 1 2  3
20 21k k 1
20! k  1!21  k !  3 
 1
k!20  k ! 20! 2

11. Ck 2  3
20 20k k
1
Ck 1 2  3
20 21k k 1
20! k  1!21  k !  3 
 1
k!20  k ! 20! 2
21  k 3
 1
k 2

12. Ck 2  3
20 20k k
1
Ck 1 2  3
20 21k k 1
20! k  1!21  k !  3 
 1
k!20  k ! 20! 2
21  k 3
 1
k 2
63  3k  2k

13. Ck 2  3
20 20k k
1
Ck 1 2  3
20 21k k 1
20! k  1!21  k !  3 
 1
k!20  k ! 20! 2
21  k 3
 1
k 2
63  3k  2k
 5k  63
63
k
5

14. Ck 2  3
20 20k k
1
Ck 1 2  3
20 21k k 1
20! k  1!21  k ! 3
  1
k!20  k ! 20! 2
21  k 3
 1
k 2
63  3k  2k
 5k  63
63
k
5
k  12

15. Ck 2  3
20 20k k
1
Ck 1 2  3
20 21k k 1
20! k  1!21  k ! 3
  1
k!20  k ! 20! 2
21  k 3
 1
k 2
63  3k  2k
 5k  63
63
k
5
k  12
T13  20C12 28312 is the greatest coefficient

16. 1
ii  Find the greatest term of 3x  4
15
when x 
2

17. 1
ii  Find the greatest term of 3x  4 when x 
15
15k 2
Tk 1 15Ck   4 
3 k

2

18. 1
ii  Find the greatest term of 3x  4 when x 
15
15k 2
Tk 1 15Ck   4  (Ignore the negative as only
3 k

2 concerned with magnitude)

19. 1
ii  Find the greatest term of 3x  4 when x 
15
15k 2
Tk 1 15Ck   4  (Ignore the negative as only
3 k

2 concerned with magnitude)
16k
 3  4 k 1
Tk  Ck 1  
15
 2

20. 1
ii  Find the greatest term of 3x  4 when x 
15
15k 2
Tk 1 15Ck   4  (Ignore the negative as only
3 k

2 concerned with magnitude)
16k
 3  4 k 1
Tk  Ck 1  
15
 2
If Tk 1  Tk then Tk 1 is the greatest term

21. 1
ii  Find the greatest term of 3x  4 when x 
15
15k 2
Tk 1 15Ck   4  (Ignore the negative as only
3 k

2 concerned with magnitude)
16k
 3  4 k 1
Tk  Ck 1  
15
 2
If Tk 1  Tk then Tk 1 is the greatest term
15 k 16k
Ck   4  15Ck 1   4 
3 k 3 k 1
 
15
2 2

22. 1
ii  Find the greatest term of 3x  4 when x 
15
15k 2
Tk 1 15Ck   4  (Ignore the negative as only
3 k

2 concerned with magnitude)
16k
 3  4 k 1
Tk  Ck 1  
15
 2
If Tk 1  Tk then Tk 1 is the greatest term
15 k 16k
Ck   4  15Ck 1   4 
3 k 3 k 1
 
15
2 2
15 k
C k   4 
3 k
 
15
 2 1
16k
15  3  4 k 1
Ck 1  
 2

23. 1
ii  Find the greatest term of 3x  4 when x 
15
15k 2
Tk 1 15Ck   4  (Ignore the negative as only
3 k

2 concerned with magnitude)
16k
 3  4 k 1
Tk  Ck 1  
15
 2
If Tk 1  Tk then Tk 1 is the greatest term
15 k 16k
Ck   4  15Ck 1   4 
3 k 3 k 1
 
15
2 2
15 k
C k   4 
3 k
 
15
 2 1
16k
15  3  4 k 1
Ck 1  
 2
15! k  1!16  k ! 4
  1
k!15  k ! 15! 3
2

24. 15! k  1!16  k ! 4
  1
k!15  k ! 15! 3
2

25. 15! k  1!16  k ! 4
  1
k!15  k ! 15! 3
2
16  k 8
 1
k 3

26. 15! k  1!16  k ! 4
  1
k!15  k ! 15! 3
2
16  k 8
 1
k 3
128  8k  3k

27. 15! k  1!16  k ! 4
  1
k!15  k ! 15! 3
2
16  k 8
 1
k 3
128  8k  3k
 11k  128
128
k
11

28. 15! k  1!16  k ! 4
  1
k!15  k ! 15! 3
2
16  k 8
 1
k 3
128  8k  3k
 11k  128
128
k
11
k  11

29. 15! k  1!16  k ! 4
  1
k!15  k ! 15! 3
2
16  k 8
 1
k 3
128  8k  3k
 11k  128
128
k
11
k  11
4
T12 15C11   4
3  11

 2

30. 15! k  1!16  k ! 4
  1
k!15  k ! 15! 3
2
16  k 8
 1
k 3
128  8k  3k
 11k  128
128
k
11
k  11
4
T12 15C11   4
3  11

 2
T12 15C11 34 218 is the greatest term

31. 15! k  1!16  k !  4 
 1
k!15  k ! 15! 3
2
16  k 8
 1
k 3
128  8k  3k Exercise 5E; 1 to 5,
 11k  128 6ac, 7bd, 8b, 10
128
k
11
k  11
4
T12 15C11   4
3  11

 2
T12 15C11 34 218 is the greatest term