then the coefficient of the divifor measures that of all the Terms of the Dividend, and all the letters of the divifor are found in every Term of the Dividend. Rule. The letter or letters in the divifor are to be expunged out of each term in the dividend, and the coefficients of each term to be divided by the coefficient of the divifor; the quantity refulting is the quotient. Ex. a) ab(b. 2aab) 6a3bc—4aˆbdm (3ac-2dm. The reason of this is evident from the nature of divifion, and from def. 6. Note. It is obvious, from corollary to prob. 3. that powers of the fame root are divided by fubtracting their exponents. Thus, a2)a3(a. a3)a7(a4. also a2b)a3bab3. CASE II. When the divifor is fimple but not a factor of the dividend. Rule. The quotient is expressed by a fraction, according to def. 8. viz. by placing the the dividend above a line, and the divifor below it. Thus, the quotient of 3ab2 divided by 2mbc is the fraction: 3ab2 2mbc. Such expreffions of quotients may often be reduced to a more fimple form, as shall be explained in the fecond part of this chapter. CASE III. When the divifor is compound. 1 RULE. 1. The terms of the dividend are to be ranged according to the powers of fome one of its letters; and thofe of the divifor according to the powers of the fame letter. Thus, if a2+2ab+b2 is the dividend, and a+b the divifor, they are ranged according to the powers of a. 2. The first term of the dividend is to be divided by the first term of the divifor, (obferving ferving the general rule for the figns ;) and this quotient being fet down as a part of the quotient wanted, is to be multiplied by the whole divifor, and the product fubtracted from the dividend. If nothing remain, the divifion is finished: the remainder, when there is any, is a new dividend. Thus, in the preceding example, a2 divided by a, gives a, which is the first part of the quotient wanted: and the product of this part by the whole divisor a+b, viz. a2+ab being fubtracted from the given dividend, there remains in this example ab+b2. 3. Divide the firft term of this new divi dend by the firft term of the divifor as before, and join the quotient to the part already found, with its proper fign: then multiply the whole divifor by this part of the quotient, and fubtract the product from the new dividend: and thus the operation is to be continued, till no remainder is left, or till it appear that there will albe a remainder. ways Thus, Thus, in the preceding example; +ab, the firft term of the new dividend divided by a, gives b; the product of which, multiplied by a+b, being fubtracted from abb, nothing remains, and a+b is the true quotient. The entire operation is as follows. a + ab ab+b2 ab+b2 ** 3a—b)3 a3—12a3 —a2b+10ab——2b2 (a2—4a+2b It often happens, as in the last example, that there is ftill a remainder from which the operation may be continued without end. This expreffion of a quotient is called an infinite feries; the nature of which shall be confidered afterwards. By comparing a few of the first terms, the law of the feries may be discovered, by which, without any more division, it may be continued to any number of terms wanted. Of the General Rule. The reason of the different parts of this rule is evident; for, in the course of the operation, all the terms of the quotient obtained by it are multiplied by all the terms of the divifor, and the products are fucceffively fubtracted from the dividend, till nothing remain: that, therefore, from the nature of divifion, must be the true quotient. Note. The fignis fometimes used to express the quotient of two quantities, between which it is placed: Thus, a2+x2÷a+x, expreffes the quotient of a2x2 divided by a+x. |