282 



THE POPULAR EDUCATOR. 



LESSONS IN ALGEBRA. XXI. 



BINOMIAL THEOREM. 



189. To involve a binomial to a high power by actual multi- 

 plication is a long and tedious process. A much easier and more 

 expeditious way to obtain the required power, is by means of 

 what is called the Binomial Theorem. This ingenious and 

 beautiful method was invented by Sir Isaac Newton, and was 

 deemed of so great importance to mathematical investigation, 

 that it was inscribed on his monument in Westminster Abbey. 



To illustrate this thereon, let the pupil involve the binomial 

 a -f- 6, and the residual a b, to the 2nd, 3rd, and 4th powers. 



Thus, (a+ 6) 2 = a 2 + 2ab + b 2 . 



(a + b) 3 = a 3 + 3a*b + 3ab 2 + b 3 . 



(a -f b) 4 a* -f 4a 3 b -f Ga?b- + 4a&' + b 4 . 

 Also, (a b)' 2 = a? - 2ab + b-. 



(a b) 3 =a?- 3, 2 & -f 3ab 2 b 3 . 



(a Ip = a 4 4a 3 6 -f Qa-b 2 4aZ> 3 + b 4 . 



By a careful inspection of the several parts of the preceding 

 operation, the following particulars will be observed to bo 

 applicable to each power, especially if carried out to a greater 

 number of powers. 



1. By counting the terms, it will be found that their number 

 in each power, is greater by 1 than the index of that power ; 

 thus, in the 3rd power the number of terms is 4 ; in the 4th 

 power it is 5, and so on. 



2. If we examine the signs, we shall perceive, when both 

 terms of the binomial are positive, that all the signs in every 

 power are -f- ; but when the quantity is a residual, all the odd 

 terms, reckoning from the left, have the sign +, and all the 

 even terms have the sign . Thus in the 4th power, the signs 

 of the first, third, and jifth terms are 4-, while those of the 

 second and. fourth are . 



3. As to the indices, it will be seen that the index of the first 

 term, or the leading quantity* in each power, always begins with 

 the index of the proposed power, and decreases by 1 in each suc- 

 cessive term towards the right, till we come to the last term, 

 from which the letter itself is excluded. Thus in (a. -{- b)* the 

 indices of the leading quantity a are 4, 3, 2, 1. 



4. The index of the following quantity begins with 1 in the 

 second term, and increases regularly by 1 to the last term, 

 whose index, like that of the first, is the index of the required 

 power. Thus, in (a -)- b)* the indices of the following quantity 

 b are 1, 2, 3; 4. 



5. We also perceive that the sum of the indices is the same 

 in each term of any given power ; arid this sum is equal to the 

 index of that power. Thus, the sum of the indices in each of the 

 terms of the 4th power is 4. 



6. As to the co-efficients of the several terms, that of the first 

 and last terms in each power is 1 ; the co-efficient of the second 

 and next to the last terms is the index of the required power. 

 Thus, in the 3rd power, the co-efficient of the second and next 

 to the last terms is 3 ; and in the same terms in the 4th power, 

 it is 4, etc. 



It is to be observed, also, that the co-efficients increase in a 

 regular manner through the first half of the terms, and then 

 decrease at the same rate through the last half. Thus, 



In the 4th power they are 1, 

 In the 6th power they are 1, 6, 



4, 6, 4, 

 15, 20, 15, 



1. 



7. The co-efficients of any two terms equally distant from the 

 extremes, are equal to each other. Thus, in the 4th power, the 

 second co-efficient from each extreme is 4 ; in the 6th power, 

 the second co-efficient from each extreme is 6 ; and the third 

 is 15. 



8. The sum of all the co-efficients in each power is equal to 

 the number 2 raised to that power. Thus, (2) 4 = 16 ; also, the 

 eum of the co- efficients in the 4th power is 16, and (2) 6 = 64 ; 

 BO the sum of the co-efficients in the 6th power is 64. 



190. If we involve any other binomial, or residual, to any 

 required power whatever, we shall find the foregoing principles 

 ti-ue in all cases, and applicable to all examples. Hence we may 

 safely conclude that they are universal principles, and may be 

 employed in raising all binomials to any required power. 



* The first letter of a binomial is called the leading quantity, and the 

 other the fallowing quantity. 



They are the basis or elements of what is called the Binomial 

 Theorem. 



The Binomial Theorem may be, therefore, defined a general 

 method of involving binomial quantities to any proposed power. It 

 is comprised in the following general rule : 



1. SIGNS. If both terms of the binomial have the sign -f-, all 

 the signs in every power ivill be + ; but if the given quantity is a 

 residual, all the odd terms in each poiver, reckoning from the left, 

 uill have the sign -f-, and the even terms 



2. INDICES. The index of the first term or leading quantity 

 must always be the index of the required power ; and this decreases 

 regularly by 1 through the other terms. The index of the following 

 quantity begins ivith 1 in the second term, and increases regularly 

 by 1 through the others. 



3. CO-EFFICIENTS. The co-efficient of the first term is 1 ; that 

 of the second ir equal to tlie index of the power ; and, universally, 

 if the co-efficient of any term be multiplied by the index of the 

 leading quantity in that term, and divided by the index of the 

 following quantity increased by 1, it 'will give the co-efficient of 

 the succeeding term. 



4. NUMBER OP TERMS. The number of terms will always be 

 one greater than the poiver required. 



In algebraic characters, the theorem is expressed thus 



1, 6, 



1. 



7l _2 '-" ' 



3" ^ 



It is here supposed that the terms of the binomial have no 

 other co-efficients or exponents thanl; but other binomials may 

 be reduced to this form by substitution. 



EXAMPLES. 



1 . What is the 6th power of x + y 



Hero, the terms without the co-efficients are x 6 , x 5 y, z*y, a 3 

 x-y 4 , ,r;/ 5 , if'. And the co-efficients, by the rule, are 

 6X5 15 X 4 20 X 3 

 v,S r' 3 4 



or 1, 6, 15, 20, 15, 6, 1. 



Now, prefixing these co-efficients to the several terms, and 

 observing the rule of. signs, we have the power required as 

 follows : 



2. What is the 5th power of x* -f 3y- ? 



Here, substituting -w for a; 2 , and b for 3y*, we have (a. -}- b) 5 

 = a 5 + 5a 4 6 + 10a 3 6 e + 10a 2 6 3 + 5ab 4 + l\ 



And restoring the values of a and b, we have (x z + 3y 2 ) 4 = 



191. When one of the terms of a binomial is a unit, it is 

 generally omitted in the 'power, except in the first or last term -, 

 because every power o-f 1 is 1 ; and this, when it is a factor, has 

 no effect upon the quantity witli which it is connected. 



EXAMPLE. Find the cube of (x + 1). Ans. x 3 + 3x" X 1 + 3x 

 X I 2 + I 3 , or x 3 + 3a 2 + 3;c + 1. 



192. The insertion of the powers of 1 is of no use, unless it 

 be to preserve the exponents of both the leading and the follow- 

 ing quantity in each term, for the purpose of finding the co- 

 efficients. But this will be unnecessary if we bear in mind 

 that the sum of the two exponents in each term is equal to the 

 index of the power. So that, if we have the exponent of the 

 leading quantity, we may know that of the following quantity, 

 and vice -versa. 



193. The binomial theorem may also be applied to quantities 

 consisting of more than two terms. By substitution, several 

 terms may be reduced to two ; and when the compound expres- 

 sions arc restored, such of them.' as have exponents may be 

 separately expanded. 



EXAMPLE. What is the cube of a -\- b -f c ? 



Here, substituting h for (b -j- c), we have a 4- (b + c) = a -{- h. 

 And, by the theorem, (a + h) 3 = a 3 + 3a~h + 3ah" + h 3 . 



Now, restoring the value of h, we have (a + 6 -f- c) 3 = a 3 + 3a 2 

 X (b -f c) + 3a X (b -f c) 2 + (b + c) 3 . 



The last two terms contain powers of (b -f- c) ; but these may 

 be separately involved, and the whole expanded. 



194. Binomials, in which one of the terms is a fraction, may 

 be involved by actual multiplication, or by reducing the given 

 quantity to an improper fraction, and then involving the frac- 

 tion. It may also be done by substitution. 



