CHAPTER X. 



BINOMIAL THKORKM. 



1. The Binomial Theorem enables us to find any power of a 

 binomial without the labor of obtaining the previous powers. In 

 order to observe the law of formation of a power of a binomial we 

 first observe the law of formation of the product of several binomial 

 factors of the form x-^a, x-\-b, x-\-c, etc., and we will afterwards 

 be able to arrive at the power of a binomial by the supposition 

 that a=b=c, etc. 



2. Law of thk Product of Factors of the form x-\-a, 

 x+b, x+c, Ktc. 



By actual multiplication it is seen that 

 (x-^a)(x+b)=x'-^(a-\-b)x-j-ab, 



(x-{-a)(x-\-b)(x-\-c)=x^-\-(a-\-b-\-c)x''-]r(ab-\-ac-\-bc)x-\-abc, 

 (x^a)(x+b)(x+c)(x-\-d)=x'-{-(a + b+c+d)x^ + 



(ab-\- ac-\-ad-\- bc-\- bd-{- cd)x- -f- (abc-\-abd-\- acd-\- bed) x-\- abed. 

 By a careful inspection of these products we will discover the 

 presence of two uniform laws — a law for the exponents and a law 

 for the coefficients. 



The law of the exponents is readily seen to be as follows : 

 The expofient of x in the first term of the product is equal to the 

 number of binomial factors, a?id in the remining terms it continually 

 decreases by 07ie until it is zero. 



The law of the coefficients may be stated thus : 

 The coeffcient of the first term is unity; the coefiieient of the sewfid 

 term is the sum of the second terms of the binomial factors: the coef- 

 ficieyit of the third term is the sum of all their different products 

 taken two at a time; the coefficient of the fourth term is the sum of all 

 their different products taken three at a time, and so on. The last 

 term is the product of all the second ter?ns of the binomial factors. 



3. Proof that the I^aw^s are General. We will now show 

 that if the laws observed above hold in the product of a given 

 number of binomial factors, they will hold in the product of any 

 number of binomial factors whatever. 



