quotient ought to have the sign+, and if they have contrary signs, the quotient ought to have the sign —; this is the rule for the signs. The individual parts of the operation are performed by the rule for the division of simple quantities. We divide the coefficient of the dividend by that of the divisor; this is the rule for the coefficients. We write in the quotient the letters common to the dividend and divisor with an exponent equal to the difference of the exponents of these letters in the two terms, and the letters which belong only to the dividend; these are the rules for the letters and exponents. 43. To apply these rules to the quantities, 5 a7 − 22 a b + 12 a5 b2 -6a4b3 - - 4 a3 b1 + 8 a2 b3, which have been employed as an example above, we place them as we place the dividend and divisor in arithmetic. Dividend. 5a7-22ab+12a5b26ab34a3b4+8a2b5|5α*_2a3b+4a2b2 -5a7+2ab — 4 a5 b2 Divisor. Quotient. a3 ·4a2 b+2b3 Rem.-20a6b+8a5 b2 — 6a4b3 —4a3b*+8a2b3 +20a6b-8a5b2+16a4b3 The sign of the first term 5 a7 of the dividend being the same as that of 5 a*, the first term of the divisor, the sign of the quotient must be +, but as it is the first term, the sign is omitted. By dividing 5 a7 by 5 a*, we have for the quotient a3, which we write under the divisor. Multiplying successively the three terms of the divisor by the first term a3 of the quotient, and writing the products under the corresponding terms of the dividend, the signs being changed to denote their subtraction (20), we have the quantity 5a2 + 2a b -4 a5 b2, which with the dividend being reduced, we obtain for a remainder ·20 a b+8 a5 b2. 6a4b3 · 4 a3 b1 + 8 a2 b3. - - By continuing the division with this remainder, the first term 20 ab, divided by 5 a*, will give for a quotient 4 a2 b, this quotient having the sign, as the dividend and divisor have differ ent signs. Multiplying it by all the terms of the divisor and changing the signs, we obtain the quantity which taken with the dividend and reduced, gives for a remainder +10a4b3-4 as b* +8 a2 b3. Dividing the first term of this new dividend, 10 a b3, by the first term, 5 a*, of the divisor, and multiplying the whole divisor by the result + 2 b3, writing the products under the dividend, the signs being changed and making the reduction, we find that nothing remains, which shows that + 2 b3 is the last term of the quotient sought. The quotient therefore has for its expression a3-4a2 b+263. 44. It is proper to remark here, that in division, the multiplication of the different terms of the quotient by the divisor often produces terms that are not to be found in the dividend, and which it is necessary to divide by the first term of the divisor. These terms are such as destroy themselves, since the dividend has been formed by the multiplication of the two factors the quotient and the divisor. See a remarkable example of these reductions; Let a3b3 be divided by a- -b. The first term a3 of the dividend, divided by the first term a of the divisor, gives for the quotient a2; multiplying this quotient by the divisor, and changing the signs of the products, we have -a3a2 b; the first term - a3 destroys the first term of the dividend, but there remains the term a2 b, which is not found at first in the dividend. As it contains the letter a, we can divide it by the first term of the divisor, and obtain + a b. Multiplying this quotient by the divisor, and changing the signs of the products we have -a2 bab2; the term-a2 b cancels the one above it, but there remains the term +ab2, which is not in the dividend. This being divided by a gives for the quotient + b2; multiplying this quotient by the divisor and changing the signs, we have ab2+b3; the first term a b2 destroys the first 3. The mechanical part of the operation will be better understood, if we look for a moment at the multiplication of the quotient a2+ab+b2 by the divisor a b. We see that all the terms reproduced in the process of dividing are those which destroy each other in the result of the multiplication. 45. It sometimes happens that the quantity with reference to which the arrangement is made, has the same power in several terms both of the dividend and divisor. In this case, the terms should be written in the same column, one under the other, the remaining ones being disposed with reference to another letter. Let there be — a1 b2 + b2 c4 — a2 c1— a® + 2 a* c2 + b® + 2ba c2 + a2 ba, to be divided by a2 -b2-c2. Arranging the first of these quantities with reference to the letter a, we place in the same column the terms a4 b2 and +2 ac2, in another, the terms + a2 b+ and -a2 c4; and in the last column the three terms + b3, +2 b* c2, + b2 c4, disposing them with reference to the letter b, as may be seen in the next page. The first term a of the dividend being divided by the first term a2 of the divisor, gives for the first term of the quotient -a; forming the products of this quotient by all the terms of the divisor, changing the signs of the products in order to subtract them from the dividend, and placing in the same column the terms containing the same power of a, we have, after the reduction of similar terms, the first remainder, which we take for the second dividend. The first term 2a4b2 of this new dividend, being divided by a2, gives for the second term of the quotient-2 a2 b2; forming the products of this quotient by all the terms of the diviso", changing the signs of the products to indicate their subtraction from the dividend, and placing in the same column the terms containing the same power of a, we have after the reduction of similar terms, the second remainder, which we take for the third dividend. The operation being continued in the same manner with the second remainder and the following ones, we shall have three terms in the quotient. The last being multiplied by all the terms of the divisor, furnishes products which being subtracted from the fourth remainder exhaust it entirely. As the division admits of being exactly performed, it follows, that the divisor is a factor of the dividend 46. The form under which a quantity appears, will sometimes. immediately suggest the factors into which it may be decomposed. If we have, for example, to be divided by 2a3 b2+1; as the divisor forms the three last terms of the dividend, it is only necessary to see if it is a factor of the three first; but these have obviously for a common factor 4a3, for 8 a-4 a3 b2 + 4 a3 = 4 a3 (2 a3 — b2 + 1). The dividend then may be represented by or 4 a3 (2 a3 — b3 + 1) + 2 a3 — b2 +1, (2 a3 — b2 + 1) (4 a3 + 1). The division is performed at once by suppressing the factor 2 a3 — b2 + 1, equal to the divisor, and the quotient will be 4a3 +1. After a little practice, methods of this kind will readily occur, by which algebraic operations are abridged. By frequent exercise in examples of this kind, the resolution of a quantity into its factors is at length easily performed; and it is often rendered very conspicuous, when, instead of performing the operations represented, they are only indicated. Of algebraic fractions. 47. WHEN We apply the rules of algebraic division to quan tities, of which the one is not a factor of the other, we perceive the impossibility of performing it, since in the course of the operation we arrive at a remainder, the first term of which is not divisible by that of the divisor. See an example; The first term ab2 of the second remainder cannot be divided by a2, the first term of the divisor, so that the process is arrested at this point. We can however, as in arithmetic, annex to the quotient a+b the fraction - ab2 + b3 a2 + b2 -, having the remainder for the numerator, and the divisor for the denominator; and the quotient will be It is evident, that the division must cease, when we come to a remainder, the first term of which does not contain the letter with reference to which the terms are arranged, or to a power inferior to that of the same letter in the first term of the divisor. 48. When the algebraic division of the two quantities cannot be performed, the expression of the quotient remains indicated under the form of a fraction, having the dividend for the numerator, and the divisor for the denominator; and to abridge it as |