524 ALGEBRA [FUNDAMENTAL IV. But it is true for 2, . . for 3, . . for 4, &c. Ex. 3. To prove the inequality, (x + y + z) B < 3 n -x" + y n + s") . From the second example of inequalities we get at once (-r + 2/ + 2) 2 <3(# 2 + 2/ 2 + * 2 ). Let us assume that (x + y + z) m <3 m ~ 1 (x m + y m + z m ), then by multiplication we get (x + y + z) m+1 < 3 m ~ : (x m+1 + y m+1 + 2 m+1 + x n y + y m x + x m z + z m x + y m z + z m y). Now, inequality, example 3, gives /. x m y + y m x + x m z + z m x + y n z + z m y< 2(x* +1 + y m+l + z " f : ), and (x + y + z) m+1 < o m (x m+l + y m+l + z m+1 ), i.e., the law is true for m+ 1, if true for m ; but it is true for 2, /. it is always true. IV. Division. 19. General Eule for the Signs. If the signs of the divisor and dividend be like, the sign of the quotient is + ; but if they be unlike, the sign of the quotient is - . This rule is derived from the general rule for the signs in multiplication, by considering that the quotient must be such a quantity as, when multiplied by the divisor, shall produce the dividend, with its proper sign. This definition of division is the same as that of a fraction ; hence the quotient arising from the division of one quantity by another may be expressed by placing the dividend above a line, and the divisor below it ; but it may also be often redu ced to a more simple form by the follow ing rules. Case 1. When the divisor is simple, and a factor of every term of the dividend. Rule. Divide the coefficient of each term of the dividend by the coefficient of the divisor, and expunge out of each term the letter or letters in the divisor: the result is the quotient. Ex. Divide lGa?xy - ZSa-xz 2 + iaPx 3 by 4a?x. The process reqiiires no explanation. It is founded on Laws II. and III., together with the rule of signs. The quotient is iay 7z 2 + x". If the divisor and dividend be powers of the same quan tity, the division will evidently be performed by subtract ing the exponent of the divisor from that of the dividend. Thus a 5 , divided by a 3 , has for a quotient a 5 " 3 = a 2 . Case 2. When the divisor is simple, but not a factor of the dividend. Rule. The quotient is expressed by a fraction, of which the numerator is the dividend, and the denominator the divisor. Thus the quotient of 3 at 2 , divided by 2mbc, is the frac- 3o& 2 tion 2mbc It will sometimes happen that the quotient found thus may be reduced to a more simple form, as shall be ex plained when we come to treat of fractions. Case 3. When the divisor is compound. Hule. The terms of the dividend are to be arranged in the order of the powers of some one of its letters, and those of the divisor according to the powers of the same letter. The operation is then carried on precisely as for division of numbers. To illustrate this rule, let it be required to divide 8a 2 + 2ab - 151 2 by 2a + 36, the operation will stand thus : -10a&-15& 2 -10t/6-15& 2 Here the terms of the divisor and dividend are arranged according to the powers of the quantity a. We now divide 8a 2 , the first term of the dividend, by 2a, the first term of the divisor ; and thus get 4a for the first term of the quotient. We next multiply the divisor by 4a, am subtract the product Sa 2 + 12a6 from the dividend; we get - Wab - 156 2 for a new dividend. By proceeding in all respects as before, we find -56 for the second term of the quotient, and no remainder: the opera tion is therefore finished, and the whole quotient is 4 a - 56. The following examples will also serve to illustrate the manner of applying the rule. Ex. 1. 3a - 6)3a 3 - 1 2a 2 - a 2 6 + 1 Oa6 - 2l-(a z - 4a + 26 -12a 2 -12a 2 + 10a& Gab - 26 2 Gab - 26 2 Ex. 2. &c. l-x + x + x 3 . Sometimes, as in this last example, the quotient will never terminate ; in such a case it may either be considered as an infinite series, the law according to which the terms are formed being in general sufficiently obvious; or the quotient may be completed as in arithmetical division, by annexing to it a fraction (with its proper sign), the numer ator of which is the remainder, and denominator the divisor Thus the completed quotient, in last example, is cc 3 1 + x + x- + 1 X If x be small compared with unity, the remainders, as we advance, continually become smaller and smaller. If, on the other hand, x be large compared with unity, the re mainders continually become larger and larger. In this case the quotient is worthless. To obtain a quotient which shall be of any practical value, we must reverse the order of arrangement, putting - x + I in place of 1 - x. The division then becomes 1 1 7, ~ &C. x x- As it is generally the largest of the quantities that we desire to divide out, we observe that, in order to effect this, we have had to begin with that quantity. Hence the Rule The terms of the divisor and dividend are to be arranged according to the powers of that letter which it is wished
(if possible) to divide out.
