Mi 



MATHEMATICS. ALGEBRA. 



1>1\ 



7, iii every term. And. generally, the sum of tho ex- 

 ponent*, in each term, it always equal to tbo exponent 



There are one or two other thin',-* which it is im- 

 I portant to take note of in looking at tlie foregoing deve- 

 nta. 



1. Tho number of term* is always one morn than the 

 number which marks tho power of the binomial. Thus, 

 when the exponent U 1, tho number of terms is two; 

 when tho exponent U 2, the number of terms is tl\ne, 

 and so on ; /. when tho exponent U odd, tho number of 

 terms is n; and when tho exponent is even, tho num- 

 ber of terms is odd. 



You cannot, therefore, with propriety, speak of a 

 middle term, except when the exponent of the power is 

 MVM ; when it is odd, there are (wo middle terms ; and 

 you see that the coefficient* of these middle terms are 

 always the tame another remarkable circumstance. 



2. But perhaps the most remarkable thing of all is. 

 that when, in the case of an cttn power, you have reached 

 the middle coefficient, as above directed, the remaining 

 coefficients are got by simply writing those which precede 

 the middle one in reverse order; so that these remaining 

 coefficients require no computing. When, in the case of 

 an odd power, yon have reached the first of the two 

 middle coefficients, you have only to repeat this coeffi- 

 cient, and then to writo all tho coefficients, before the 

 middle ones, in reverse order, as in the other case ; so 

 that no coefficient beyond the middle one, or the middle 

 ]><iir, need ever be computed; they have only to be 



;'. 



The present examination affords a good opportunity to 

 bring another particular for our attention. Tho fore- 

 going table presents us with a set of equations, but they 

 differ from the equations solved at pages 450, &c., in a 

 very marked manner; you must take notice of this. 

 The equations at the pages referred to fix certain condi- 

 tion!, which tho value of x must be such as to satisfy 

 no other values of z would do ; but in the equations in 

 tho preceding table, no condition* are implied both a 

 and x in each may stand for anything whatever ; for the 

 second side, or member, is only the first side developed, 

 unfolded, or spread out ; in other words, one side is 

 nothing but the other side put in a different form ; so 

 that, write what we will for o and r, the two sides must 

 of necessity remain equal. Equations such as these, in 

 which one side is only the other iu a changed form, are 

 called identical equations, or simply identities. Now, as 

 in identities anything may be substituted for the letters, 

 yon may in the table put y for x ; you will thus get 

 the developments of the different powers of a y, or, to 

 be uniform, of a x ; all you have to do is to change x 

 into x on both sides ; you will see that tho only change 

 in the development is that the second, and every alternate 

 term, becomes minus ; thus 



(a x)' a 3 3a'x + Sax'x* 

 1 



and so on. With these changes, therefore, in the alter- 

 nate signs, the table exhibits the developments of the 

 power both of o+x and o x. 



\Yh:it has now been said in reference to the powers 

 of a-\-r, that is, in reference to tho powers of a plus or 

 MMIIU r, must bo very carefully read over, and fully 

 understood. Wo have been giving to you the leading 

 particulars of tho. celebrated BINOMIAL THEOREM, and 

 >>! must try to impress them on your mind. In order 

 to this, you should writo out the developments of (a r) 1 , 

 (a *), *c., and to try (a+x)*, (a z), &c., finding 

 the coefficients according to the short and easy method 

 explained above ; that is, deriving them, 0110 after 

 ;,i...ti,. r, frin the law shown to prevail, as far at least as 

 to tho eighth power. It prevails universally, but the 

 nl demonstration of tho Iliiuimial Theorem reqnin s 

 advanced principles of Algebra ; you will do right 

 to refuse assent to the law at present, for an exponent 

 higher than 8, unless yon like to put it to tho test for 

 9, 10, Ac. You will find the theorem in a more general 

 form hereafter. As an application of the Binomial 



Theorem, lot it bo required to developo (a 3y) 5 , which 

 is the tamo as (a-f-x)*, when 3y U put for z. 



.11 what is shown above, it appears that the terms, 

 without the coefficient, are 



a", a (Zy), a (:{y), a* (3y), a (3y), (3y). 

 And from the law of tho coefficients, these are 



1, 5, LJ.5 (or 10), 10, 5, 1 ; 



the hut three being those of the first three written in 

 reverse order ; therefore, remembering to write tho terms 

 alternately plus and minus, 



Again, let Example 2, page 461. be taken ; namely, 

 (3oj 4y) 3 . The terms, without the coefficients, are 



(So*)', <3ox)4y, 3<u-(4-,)=, (4y); 



therefore, introducing the coefficients, 1, 3, 3, 1, wo 

 have 



(3o 4y) = (3a*)S 3 (3ox)4./ + 3.3ax (4y)= (4y) 



2 y + 144axy a G4y. 



i:\ AMi'Li's rui: i.\i:i.r, j;. 



. . 



6. (x + ty*)'- 

 9. (x 2y*)'. 



2. (a*) 5 . 3. 



4. (1 x)'. b. (l-f-3x). 



7. (* a + 3y ! )'. 8. (2a *). 

 NOTE. The preceding process may be extended to 

 expressions of three terms, four terms, <tc., in tho 

 manner following : 



[a+ 6 4 c] =] ( 



c] = (a 4 ) 4 J ( + >) + 

 M 



= o 4* 4<* +2 (all +<u 4c) 



[o 4 6 4 e 4rf] = [ (a +b) + tc+d) ]=(o4) 4 J (o H) (4<*) 4 (c4rfl 

 = o 4 ai 4 4 ! 4 ad 4 fc> 4 M) +t*+tcd+i* 



(o 4 



So that the square of the sum of three quantities, or of 

 the sum of fuitr quantities, is equal to the squares of 

 the quantities themselves, together with twice the sum 

 of tlio products of every possible pair of them ; and 

 the same is truo for Jive, and for any number of quan- 

 tities. An extended use of involution will bo found 

 in the calculation of logarithms, in a subsequent chap- 

 ter. 



v 



DIVISION. 



When one quantity (called the dividend) is to be 

 divided by another (called the divisor), the object is to 

 find a third quantity (called the quotient), such, that if 

 the quotient and divisor bo multiplied together, the 

 product will bo the dividend ; and this is to be brought 

 about whether the quantities concerned are tho figures 

 of arithmetic, or the symbols of algebra. Tho rule for 

 division is thus suggested by that for multiplication ; it 

 is as follows : 



CASE I. When dividend and divisor are both simple 

 quantities. 



RULE 1. Determine tho sign of the quotient, on tho 

 principle that if the signs of dividend and divisor be 

 like, the quotient is plus : if unlike, minus. 



2. Having found the sign, next find the coefficient by 

 dividing tho coefficient of the dividend by that of tho 

 divisor, as in common arithmetic. 



3. To the coefficient, annex the letters, which must bo 

 such, that when they are united to the letters ah. 



in the divisor, they may exactly make up those iu tho 

 dividend 



It is plain, that by following those precepts, you will 

 get a quantity for quotient such, that when it and tho 

 divisor are multiplied together, tho resulting sign will 

 be that of the dividend, the resulting coefficient also 

 that of the dividend, and finally the resulting letters, 

 arising from combining those of divisor and quotient 

 together, will be the same as the letters in the dividend. 



If it should happen that there are letters in the 

 divisor, that is to say factors in the divisor, which are 

 not also in the dividend ; then, like as in arithmetic, 

 since actual division by those cannot bo performed, tliry 

 must bo written as divisors in the quotient ; the quotient 

 will then appear as & fraction incapable of reduction to a 

 simpler form just as in division of numbers. 



