134 Algebra. 



that IS, the parenthesis becomes a\ so that the 



fourth term itself becomes 



1.2.3 ' 



and so on for the other terms. 



The last term reduces to the product oi 11 a's ; that is, to 



a" . 

 Therefore, on the supposition that a=a=a^^= . . . =a„, the 

 equation above written becomes 



(x — a)"= 



1.2 1.2.3 



which is the Binomial Formula. 



The expression on the right-hand side of the equation is called 

 the Expansion or the Development of the power of the binomial. 



5. Example. Expand (y+2)^. 



Substitute r for x, 2 for a, and 5 for 71, in the binomial formula 

 and we obtain 



1.2 ' 1.2.3 ' T.2.3.4 



or simplifying, 



(y + 2 )'^=y^ + I oy -f 40_>/^ + Soy^ + 8oj/ +32. 



6. Binomial Theorem. The binomial formula may be stated 

 in the form of a theorem as follows : 



hi any poiver of a binoinial x-\-a, the exporient of x begins in the 



first term with the exponent of the poiver, and in the following terms 



continually decreases by one. The expojient of a commences ivith 



one in the second term of the power, arid conti?iually increases by one. 



The coefficient of the first tei'^n is o?ie, that of the second is the ex- 

 pone?it of the power ; arid if the coefficiejit of any term be multiplied 

 by the exponent of x i7i that ter7n a7id divided by the expo7ient of a 

 i7icr eased by 07ie, it 7v ill give the coefficie7it of the succeeding terr7i. 



7- HiSTOKiCAii Note. The first rule for obtaining the powers of a bi- 

 nomial seems to have b(?en given by Vieta (1540-l(i03). He observed as a 

 necessary result of the process of multiplication that the successive coefficients 

 of any power of a binomial are : first, imity ; second, the sum of the first and 



