BINOMIAL. 



iX i i*3+,fcc. = 



b 1 63 



_ _-__i __ _ &c 



8a|^ 16a|' 



To investigate this theorem, suppose n 

 quantities, x + a, x + 6, x + c, &c. 

 multiplied together; it is manifest that 

 the first term of the product will be .r, 

 and that x ', x n s , &c. the other pow- 

 ers of x, will all be found in the remain- 

 ing terms, with different combinations of 

 , 6, c, </, &c. _ _ 



8cc = 



9 



3 -|- 5cc. 



This proof applies only to those cases 

 in which n is a whole positive number; 

 but the rule extends to those cases in 

 which n is negative or fractional. 



Ex 1. a + x** = a z + 8 ai a+ 28 a 6 x- 

 + 56 u5 a-3 + 70 a4 ^4 + 56 o3 x* + 28 a- 

 x 6 + 8 a x 1 + x 8 . 



. 77 1 



-f B xn 



B xn + &c. and x+x X a, 1 " 1 + P^ f ' J 



&c. or, 



i + Q.rn_i+& c .- are the 

 --axn ' -\-dPxn 2 +&,c. 3 sameseries; 

 therefore, A=P -|- a, B Q, + a P, &c. 

 that is, by introducing one factor, x+, 

 into the product, the co-efficient of the se- 

 cond term is increased by a, and by intro- 

 ducing a+6into the product, that co-effi- 

 cient is increased by b, &.c. therefore the 

 whole value of A is a + b + cH~ d -f- & c - 

 Again, by the introduction of one factor, 

 ;r+a,Jthe co-efficient of the third term, Q, 

 is inM-eased by a P, i. e. by a multiplied 

 by tire preceding value of A, or by aX 

 b -{- c-\-d-\- Sec. and the same may be 

 said with respect to the introduction of 

 every other factor; therefore, upon the 

 whole, 



B = a ."H-C-HH-&C. 



-f b . c-H-f&c. 

 + c . d-f&c. 

 In the same manner, 



C = a . b . c .+</+ & c - 

 + a.c . d+ &c. 

 f- 6 . c . f/+ 8cc. 



and so on ; that is, A is the sum of the 

 quantities a, 6, c, &c. B is the sum of the 

 products of every two ; C is the sum of the 

 products of every three, &c. &c. 



Let a = b =sz c = d = &c. then A, 

 ora + 6-r-c-}- r/ + &c. = na ; = ab -f- 

 ac B 4- be H- &c. = a 1 X the number of 

 combinations of a, 6, c, </, &c. taken two 



and two, =?i .- a 1 ; in the same manner 



, &c. 



If either term of the binomial be nega. 

 live, its odd powers will be negative, and 

 consequently the signs of the terms, in 

 which those odd powers are found, will 

 be changed. 



Ex. 4 a x 9 = a 8 8 aT 07 + 28 a 6 x- 

 56 a* x +70 a* x= 56 o3 xi + 28 a 1 xff 

 8 a xi 4- r. 



07. 5. a 1 ic^n = a 1 " TZ a 1 J j; J + 

 a;4 & c . 



If the index of the power to which a 

 binomial is to be raised be a whole posi- 

 tive number, the series will terminate, be- 



cause the co-efficient n . ^ .- ~--. 



o o 



&c. will become nothing when it is con- 

 tinued to n + 1 factors. In all other 

 cases the number of terms will be indefi- 

 nite. 



When the index is a whole positive 

 number, the co-efficients of the terms ta- 

 ken backward, from the end of the series, 

 are respectively equal to the co-efficients 

 of the corresponding terms taken for- 

 ward from the beginning. 



Thus, in the first example, where a-\-x 

 is raised to the 8th power, the co-efficients 

 are, 1, 8, 28, 56, 70, 56, 28, 8, 1. 



In general, the co-efficient of the ?* 



term is- 



.n 1. 



.2.1 



=1. 



1.2. 



n 2 . n 1 . n 



it appears that C = ?i.- 



2 



And.r + q ^-h* . a'+c . &c. to n factors 

 x - ol ; therefore x + a 



The co-efficient of the n th term is ? ' ' n 

 2....S .2. = n ; of the w _ lt h term, 



1 . n 2. . . .3. n .n 1 



n 2 



1 .2 



-,&c. 



