ing 1 from the exponent of the preceding term, which is equivalent to dividing by a or a1, (Art. 6.) a1 Thus, since the quotient obtained by dividing any quantity by itself is unity. Hence the singular result, α a1 =1, a=1, which is true, independently of the value of a; so that if a=1, 2, 3, 4, &c., we shall have, 1o=1, 2o=1, 3o=1, 4o=1, &c. CHAPTER III. BINOMIAL THEOREM. 27. The rules for involution and evolution already given apply only to monomials, or quantities consisting of a single term. Binomials, or quantities consisting of two terms, connected by the signs + or -, may be involved by actual multiplication. Thus, the powers of a+b may be found by actually multiplying it by itself, as follows: a + b a + b a2+ ab ab +b2 (a+b)= a2+2ab+bo a+b a3b+2ab+b3 (a+b)3= a3+3a3b+3ab3+b3 In this way we may proceed to find any power of a+b, but the process will evidently be very tedious for high powers. Some general rule is therefore desirable, which will enable us to express any power of a binomial without performing the multiplication. On the other hand, some rule or formula is desirable, by which we can express any root of a binomial. We have seen, in the case of monomials, that fractional and negative exponents may be treated by the same rules as positive and integral ones; and (Art. 25,) that with the use of fractional exponents the same rule effects both involution and evolution. We might infer that the same generality exists for the powers and roots of binomials, and that the same rule, or formula, by which we can express any power, will also serve to express any root. In fact, such a formula is Newton's Binomial Theorem, by which we can express the value of (a+b)", whether m be positive, negative, integral or fractional. Thus, if m is negative or =―n, this theorem enables us to express 1 the value of (a+b)−* or (Art. 20,) of (a+b)". If m = p' Va+b. INDETERMINATE COEFFICIENTS. 19 1 it expresses the value of (a+b)7, or (Art. 19,) of 1 1 it expresses the value of (a+b)P, or (Art. 19,20,) 9 P It is in these applications that the great utility of this theorem is manifested. We shall here demonstrate it in the most general manner, and in the next chapter give the applications to the particular cases. We must first establish the following important principles. INDETERMINATE COEFFICIENTS. 28. Whenever we have an equation of the form, A+Bx+Cx2+Dx3+ &c.=A'+B'x+C'x2+D'x3+ &c., which is true whatever be the value of x, then, the coefficients of the like powers of x in the two members are equal each to each, and we shall have, A=A', B=B', C=C', &c. For since the equation is true whatever be the value of x, it is true for the value, x=0. But if this value is substituted for x, all the terms become 0, except A and A', which being independent of x, that is, not containing x in any form, do not become 0. Therefore the equation is reduced to A=A'. These quantities, therefore, destroy each other in the original equation, which then becomes 1 Bx+Cx2+Dx3+ &c. =B'x+C'x2+D'x3+ &c. Divide this by x, and we have B+Cx+Dx2+ &c.=B'+C'x+D'x2+ &c. which must also be true when x=0; but in that case all the terms become 0 except B and B', and we have simply B=B'. Our equation, therefore, is Cx+Dx2+ &c. =C'x+D'x2+ &c. And it is shown in precisely the same manner, that C=C', D=D', E=E', &c. 29. It follows from this, that if we have an equation of the form A—A'+(B—B')x+(C—C')x2+(D—D')x3+ &c.=0, which is true for any value of x, the coefficients must all be equal to 0; for this equation is derived from the one proposed in the preceding article, by transposing all the terms to the left hand member, and it has been shown, that A=A', B=B', &c., whence A-A'=0, B-B'=0, &c. And if we put A-A'—M, B—B'—N, &c., the equation will be, M+Nx+Px+Qx2+ &c.=0. Whenever, then, we have an equation of the form, M+Nx+Px+Qx3+ &c.=0, which is true for any value of x, all the coefficients are equal to 0, and we shall have, M=0, N=0, P=0, &c. 30. This principle, simple as it is, is one of the most prolific in algebra. It is the distinguishing feature of algebra, that its processes are general, and embrace large classes of particular cases. Hence it is often employed to investigate formula which shall have the same form, whatever be the value of certain of the quantities which enter into them; as in the case of the binomial theorem, it is employed in discovering a formula for expressing any power or root of a+b, which formula shall be the same whatever are the values of a and b, and whatever the power or root sought. It is in such investigations that the principle above demonstrated is particularly useful. A series is assumed to be the required formula, but the coefficients are all unknown quantities, and are consequently called indeterminate coefficients. If an equality can then be shown to exist between this series and another of the same form, the above principle gives us as many equations as there are coefficients, from which the values of the coefficients are then determined by elimination. These remarks will be clearly comprehended by attending to the demonstration given in Art. 33. xm-ym 31. The difference of two powers of the same degree, xm—ym, is exactly divisible by the difference of their roots, x-y, if m is a positive integer. The process of division will be, OF THE QUOTIENT OF 2d rem. =x-2y3—ym M Now it is evident, that by continuing this process, the next term of the quotient will be found to be am-3y3, and that the 3d remainder will be xm-sy3—ym; the 4th remainder xm―1y1—ym, &c.; so that the mth remainder will be xm-my-ym. But xm―m=xo=1, (Art. 26,) therefore, this last remainder is ym-ym-0; that is, the process terminates at the mth term, since the remainder is equal to 0; and the division is consequently exact. The last term of the quotient may be obtained by considering the (m-1)th remainder, which will be xy-1-y", which divided by x-y gives ym-1. In the same way, from the (m-2)nd remainder the last term but one of the quotient is found to be xym-2. The form of the quotient is therefore evident, and we have xm-ym = =xm—1. x-y The number of terms in this quotient is equal to m; for y is contained in all the terms except the first, and the exponents of y are 1, 2, 3, 4, &c. to m-1; so that the number of terms containing y is m-1, and the whole number of terms, including the first, is m-1+1 or m. when x=y xm-ym 32. The value of x-y whether m is positive, negative, integral or fractional. The value of this quotient, when m is a positive integer, has been found in the last article to be xm-ym x-y When x=y, this equation becomes xm = =xm—1+xm—1+xm-1+.. ·+xm-1+xm-1+xm—1. XX But the number of terms in the right hand member has been shown to be m, therefore we have mx" xm is in all cases mxm-1, m ..... =mxm-1; X-X so that our proposition is true when m is a positive integer. To extend it to the other cases, FIRST, let m be a negative integer, or let m=n. Then we have ̄x—"y—"(x”—y") ̧ xm_ym___x_n_y-n__x-ny-n(yn—xn) x-y x-y x-y x-y But n is a positive integer; therefore, by what has just been shown, when x=y xn-yn Xn. -xn = x-y XX Also, x▬1y—”—x1x¬n=x-2", (Art. 5.) .2n |