Multiplying the divisor by this quotient, and subtracting the product from the dividend, I have for a remainder - 5 ab+b2, the a quantity which, according to the principle stated at the commencement of this article, must have with 4 a2 same greatest common divisor as the first. Profiting by the remarks made above, I suppress the factor b, common to all the terms of this remainder, and multiply it by 4, in order to render the first term divisible by that of the divisor; I have then for a dividend, the quantity 16 b2, and the quotient thence arising is 3. Multiplying the divisor by the quotient, and subtracting the product from the dividend, we obtain the remainder and the question is reduced to finding the greatest common divisor to this quantity, and But the letter a, with reference to which the division is made, not being in the remainder, except of the first degree, while it is of the second degree in the divisor, it is this which must be taken for the dividend, and the remainder must be made the divisor. Before beginning this new division, I expunge from the divisor 19 a b 19 b, the factor 19 b, common to both the terms, and which is not a factor of the dividend; I have then for a dividend, the quantity 4 a2 5 ab + b2, The division leaves no remainder; so that a common divisor required. By retracing these steps, we may prove à posteriori, that the quantity ab must exactly divide the two quantities proposed, and that it is the most compounded of those which will do it. In dividing by ab the two quantities proposed, a2 3a33a2 b + a b2 — b2, 4 a2 b5 a b2 + b3, we resolve them as follows; (3 a2 + b2) (a — b), (4 a b — b3) (a — b). 49. When the quantity, which we take for a divisor, contains several terms having the letter, with reference to which the arrangement is made, of the same degree, there are precautions to be used, without which the operation would not terminate. See an example of this. Let there be the quantities if we make the preparation as for common division, by dividing, first, a2 b by a b, we have for the quotient a; multiplying the divisor by this quotient, and subtracting the products from the dividend, the remainder will contain a new term, in which a will be of the second degree, namely, a2 c, arising from the product of ac by a. Thus no progress has been made; for by taking the remainder for a dividend, and multiplying by b, to render the division possible by ab, we have and the term ac produces still a term a2 c2, in which a is of the second degree. To avoid this inconvenience, it must be observed, that the divisor abac + d$ = a (b — c) + da, by uniting the terms ab- ac in one; and, for the sake of shortening the operation, making b— cm, we have for the divisor a m+ d2; but then the whole dividend must be multiplied by the factor m, to make a new dividend, the first term of which may be divided by a m, the first term of the divisor; the operation then becomes The terms involving a now disappear from the dividend, and there remain only the terms which have the first power of a. To make these disappear, we first divide the term a c2 m by a m, and it gives for a quotient c; multiplying the divisor by this quotient, and subtracting the products from the dividend, we obtain the second remainder. Taking this second remainder for a new dividend, and suppressing the factor d2, which is not a factor of the divisor, we have The remainder bd2 c2 m dm2. am + d2 dm of this last division, not involving a, it follows, that if the proposed quantities have a common divisor, it is independent of the letter a. Having arrived at this point, we can continue the division no longer with reference to the letter a; but it will be observed, that if there be a common divisor, independent of a, to the quantities b d2 - c2 m dm2 and am + d2, it must divide separately the two parts am and d2 of the divisor; for if a quantity is arranged with reference to the powers of the letter a, every divisor of this quantity, independent of a, must divide separately the quantities multiplied by the different powers of this letter. To be convinced of this, we need only observe, that, in this case, cach of the quantities proposed must be the product of a quantity depending on a, and of the common divisor, which does not depend upon it. Now ifwe have, for example, the expression A a1 + Ba3 + C2 a2 + Da + E, in which the letters A, B, C, D, E, designate any quantities whatever, independent of a, and it be multiplied by a quantity M, also independent of a, the product MA a+MB a3 + MC a2 + MD a + ME, arranged with reference to a, will contain still the same powers of a as before; but the coefficient of each of these powers will be a multiple of M. This being supposed, if we restore the quantity (b— c) in the place of m, we have the quantities bd2c2 (bc) — d (b — c)2, a (bc) + d2; and it is evident, that b c and d2 have no common factor; the two quantities then under consideration have not a common divisor. If it were not evident by mere inspection, that there is no common divisor between b c and d3, it would be necessary to seek their greatest common divisor by arranging them with reference to the same letter, and then to see if it would not also divide the quantity bd2 - c2 (b c)3. 50. Instead of putting off to the end of the operation, the investigation of the greatest common divisor independent of the letter with reference to which the quantities are arranged, it is less trouble to seek it at first, because, for the most part, the operation becomes more complicated at each step as we advance, and the process is rendered more difficult. Let there be, for example, the quantities aa b2 + a3 b3 + b2 c2 a2 b + a b2 + b3 a c2 a3 b c2 b2 c1, a2 c a b c -- b2c; having arranged them with reference to the letter a, we have I observe, in the first place, that if they have a common divisor which is independent of a, it must divide each of the quantities, multiplied by the different powers of a (49), as well as the quantities bc-b2 c4 and b3-b2 c, which do not contain this letter. The question is reduced then to finding the common divisors of the two quantities b2c2 and bc, and determining whether among these divisors there is to be found one which will divide at the same time bbc and b2 bc, b4 c2 b2 c4 and b3 — b2 c. Dividing bac2 by bc, we find an exact quotient b+c; bc then is a common divisor of the quantities b2 c2 and bc, which evidently admit of no other, since the quantity b-c is divisible only by itself and by unity. We must now see whether bc will divide the other quantities referred to above, or whether it will divide the two quantities proposed; it is found that it will, and it gives (b + c) a2 + (b2 + b c) a3 + b3 c2 + b2 c3, To bring these last expressions to the greatest degree of simplicity, we should see if the first is not divisible by b + c; it appears upon trial that it is, and we have only to find a common divisor to the quantities a1 + ba2 + b2 c2, a2 + ba + b2. By proceeding with these as the rule prescribes, we come, after the second division, to a remainder containing the letter a of the first power only; and as this remainder is not the common divisor, we conclude that the letter a does not make a part of the common divisor sought, which therefore can consist only of the factor bc. If, beside this common divisor, another had been found, involving the quantity a, it would have been necessary to multiply these two divisors together to obtain the common divisor sought. These remarks will enable the learner, after a little practice in analysis, to find in every case the greatest common divisor. He will determine without difficulty that the quantities. 9a3b-27 a2bc6abc2 + 18bc3, have for their greatest common divisor the quantity 3 a2-2 c2. 51. The four fundamental operations, addition, subtraction, multiplication and division, we perform in algebra as in arithmetic, observing merely to proceed, in the operations prescribed by the rules of arithmetic, according to the methods given for algebraic quantities. I shall, therefore, merely suggest these methods, giving an example of the application of each. I shall begin as I did in arithmetic, with the multiplication and division of fractions, as they require no preparatory transformations. 1. For multiplication, we have |