may be considered as consisting of corresponding pairs, of (6.) To illustrate what has been done, take the following EXAMPLES. 1. To find the roots of 223 3z2 3z+2=0. Since 1 is clearly one of the roots, by dividing the equation by 2+1 and putting the quotient equal to 0, we get the equation 222-52 + 2 = 0; which is a reciprocal or recurring equation, whose solution gives 2 and for the remaining roots of the given equation. 2. To find the roots of 525 + 424 + 323 322 42-50. Since 1 is one of the roots, if we divide the equation by 2-1, and put the quotient equal to 0, we shall get 52*+923+ 1222 +92 +5 = 0, or 5(22 + 1) + 9 ( 2 + 1 ) · +12=0. Putting =x, the preceding equation is easily reduced to 5x2 + 2; whose solution gives x = 9-141 and x= 10 9 + 141 = 10 + 1 2 ; whose solution will give the remaining roots of the given equation, which are all imaginary. 3. To find the roots of 24 +223 22 1 = 0. Since 1 and 1 are clearly two of the roots, if we divide the equation by 22 - 1 and put the quotient equal to 0, we shall get the reciprocal or recurring equation z2 + 2 + 1 = 0; whose solution gives 1 and 1 for the remaining roots of the given equation. SOLUTION OF BINOMIAL EQUATIONS. (1.) Agreeably to what has been said in Evolution, at p. 282, etc., we may clearly represent any binomial equation by y" ±a” = 0, (1). Putting y = ax, (1) becomes a"x" ±a" = 0, which, by rejecting the factor a”, is reduced to a2±1=0; for which we shall take the equivalent separate equations x"=-1, (2), and a" = 1, (3). we represent any positive or negative odd integer by m, and any positive or negative integer (including 0) by p, it is clear that for (2) and (3) we may write x" = + 114m and √2 and suppose m and p to be positive integers. It is clear from (4) and (5), that the values of a (generally) occur in pairs; since by using +for+ in 1-1 we get one root, and by taking for we get another corresponding root. (3.) Putting m = n ±q, multiplying by 4 and dividing by 4m n, we get =4+ which being put for the exponent n 4q n and x= (1± 1/2 4q 12 = Because the last two of these values of x are the same as the first two taken in a reverse order, it follows that the values of which correspond to m greater than n, will be merely a repetition of the values of a which correspond to m less than n. Hence, supposing n to be an odd integer, by putting 1, 3, 5,. . . . n successively, for m in (4), we shall get the n dif x= 2 pairs of values of x, together with the value of x ex will evidently be the n roots of (4), since it is clear that (4) can not have equal roots. And if n is an even number, by putting 1, 3, 5, . . . . n−1 successively, for m in (4), we shall in like manner get the n which, as before, will be the n roots of (2). (4.) If n in (5) is an odd number, and we put p= n - 1 2 √2 (1± 12 it is clear that the values of a which result of the values of a resulting from taking p not greater than in (5), we shall get the n values of x expressed by x=1, x= which will be the n roots of (3). Again, if n in (5) is an even number, and we put p= 8p 88 2 nent in (5), reduces it to x= as before, the values of a resulting from taking p greater n than will be merely a repetition of the values of a which 2 (5), we shall get the n values of x expressed by a = 1, x = (5.) It is evident that (4) and (5) may be written in the (7); consequently, having found the values of 8 √2 the n roots of (2) will be found by putting Ι 1, 3, 5, etc., successively, for m in (6), and those of (3) will be found by putting 0, 1, 2, 3, etc., successively, for p in (7), and then developing the powers. 1. to change 1 (in them) into If we convert the binomial (1 + a √-1) into a series arranged according to the ascending powers of a √ 1, we shall (by the binomial theorem) get (1 + a √ − 1)'= 1 + - etc.)+(ra . -- 1) r(r — 1) (r− 2) a3 √ — 1+ a2 - etc. (1. = 1 2 r(r−1) 3 a2+ r(r−1)(r−2) r(r-1)(r-2) (r−3) (r−4) 1.2.3 as etc.) x − 1, (8); consequently, if r is a positive integer, the power will be exact, and if r is fractional, and a sensibly less than 1, the value of (1 + a √1)" can be found to any required degree of exactness by converging series. |