Deriving the Formula for the Polynomial Series

Pascal’s Triangle

 

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

 

Descending powers with Pascal’s triangle numbers as coefficients:

 

2x

3x – 2x

4x – 2∙3x + 2x

5x – 3∙4x + 3∙3x – 2x

6x – 4∙5x + 6∙4x – 4∙3x + 2x

7x – 5∙6x + 10∙5x – 10∙4x + 5∙3x – 2x

 

3x – 2x

30 – 20 = 0

31 – 21 = 3 – 2 = 1 = 1!

 

4x – 2∙3x + 2x

40 – 2∙30 + 20 = 1 – 2 + 1 = 0

41 – 2∙31 + 21 = 4 – 6 + 2 = 0

42 – 2∙32 + 22 = 16 – 18 + 4 = 2 = 2!

 

5x – 3∙4x + 3∙3x – 2x

50 – 3∙40 + 3∙30 – 20 = 1 – 3 + 3 – 1 = 0

51 – 3∙41 + 3∙31 – 21 = 5 – 12 + 9 – 2 = 14 – 14 = 0

52 – 3∙42 + 3∙32 – 22 = 25 – 48 + 27 – 4 = 52 – 52 = 0

53 – 3∙43 + 3∙33 – 23 = 125 – 192 + 81 – 8 = 206 – 200 = 6 = 3!

 

6x – 4∙5x + 6∙4x – 4∙3x + 2x

60 – 4∙50 + 6∙40 – 4∙30 + 20 = 1 – 4 + 6 – 4 + 1 = 8 – 8 = 0

61 – 4∙51 + 6∙41 – 4∙31 + 21 = 6 – 20 + 24 – 12 + 2 = 32 – 32 = 0

62 – 4∙52 + 6∙42 – 4∙32 + 22 = 36 – 100 + 96 – 36 + 4 = 136 – 136 = 0

63 – 4∙53 + 6∙43 – 4∙33 + 23 = 216 – 500 + 384 – 108 + 8 = 608 – 608 = 0

64 – 4∙54 + 6∙44 – 4∙34 + 24 = 1296 – 2500 + 1536 – 324 + 16 = 2848 – 2824 = 24 = 4!

 

7x – 5∙6x + 10∙5x – 10∙4x + 5∙3x – 2x

70 – 5∙60 + 10∙50 – 10∙40 + 5∙30 – 20 = 1 – 5 + 10 – 10 + 5 – 1 = 0

71 – 5∙61 + 10∙51 – 10∙41 + 5∙31 – 21 = 7 – 30 + 50 – 40 + 15 – 2 = 72 – 72 = 0

72 – 5∙62 + 10∙52 – 10∙42 + 5∙32 – 22 = 49 – 180 + 250 – 160 + 45 – 4 = 344 – 344 = 0

73 – 5∙63 + 10∙53 – 10∙43 + 5∙33 – 23 = 343 – 1080 + 1250 – 640 + 135 – 8 = 1728 – 1728 = 0

74 – 5∙64 + 10∙54 – 10∙44 + 5∙34 – 24 = 2401 – 6480 + 6250 – 2560 + 405 – 16 = 9056 – 9056 = 0

75 – 5∙65 + 10∙55 – 10∙45 + 5∙35 – 25 = 16807 – 38880 + 31250 – 10240 + 1215 – 32 = 49272 – 49152 = 120 = 5!