For the rule of Letters, we must efface the similar letters in the dividend and divisor, which has no need of demonstration, since in multiplication we are satisfied with writing them one after abc α another. Thus = bed d For the rule of Exponents; when there are similar letters or quantities in the dividend and divisor, with different exponents, we must subtract the less exponent from the greater, then write this difference instead of the greater, and efface the letter, or the quantity, which is found to have the less exponent. sequence of the rule established in multiplication for what relates to exponents. What we have just said, teaches us to divide a monomial by a monomial, which would be equally applicable to the case in which we should have a polynomial to be divided by a monomial; since the operation would then be reduced, to divide separately each of the terms of the polynomial by the monomial; but if we had a polynomial to be divided by a polynomial, the following method must be pursued. We begin by arranging the dividend and divisor with respect to a same letter. The arranging a polynome consists only in writing the terms of the polynomial in such an order, that the first may contain the letter by which we arrange, raised to its highest exponent; the second may contain the same letter rai sed to an exponent next less, &c. That being performed in both numbers, we divide the first term of the dividend by the first term of the divisor, we write the quotient under the divisor; then we multiply all the divisor by the quotient, to subtract it from the dividend; we proceed in this according to the principles of division of numbers. Suppose we have 6a2b2+4ab3 +b1+a*+4a3b to be divided by 2ab+b2+a2. I arrange the dividend and divisor with respect to the same letter a, and I then write ; Ist rem. a^+4a3b+6a2b2 +4ab3 +b1 ] a2+2ab+b• -a^— 2a3b — a 2 b 2 2nd rem. 3d rem. We divided at by a2, then we multiplied all the divisor by the quotient a; we wrote all the products under the dividend, with the attention of changing their signs; we obtained a first remainder; we divided the first term of this remainder by the first term of the divisor, which gave 2ab in the quotient; we then multiplied the divisor by this new partial quotient; then, &c. Another Example, in which the two members of the division are arranged. EXAMPLE. 20a-41a+b+50a3b-45ab3+25ab6b5a3-4a2b+5ab2-36* -20a3 +16a1b-20a3b2+12a2b3 -25a+b+30a3 b2 —33a2b3 +25ab*—6b5 4a2-5ab+26* 1st rem. +25a+b-20a3b2 +25a3b3-15ab1 It often happens, that the division cannot be performed exactly; then FRACTIONS. of algebraical fractions, as in those of numbers. The same rules are to be followed in the calculation But to reduce fractions to the same denominator, there is a case, which it is proper to examine rela tively to the signs. Let it be proposed to reduce the two terms of the quantity a a- -b and a+b to the same denominator; by operating according to the principles of arithmetic, we shall have axa+b-a-b×a—d (a−d) · (a+b) effectuating the calculation, the quantity will bea2+ab-a2+ab+ad-bd We must remark, relatively to the signs, that, having made the multiplication of ab by a-d, we have changed all the signs of the product, because the sign, which is before the second term of the quan tity proposed, indicates that this second term is to be subtracted from the first, (the algebraical subtraction consisting in changing the signs of the quantity to be subtracted.) In effectuating the multiplication of ab by ad, we said + a multiplied by +a, gives a2; but as it must be subtracted, we have written2; then-6 multiplied by +a, gives ab, but as it is to be subtracted, we have written +ab, &c. In such cases, beginners are apt to make mistakes in fixing the signs: Let them consider, that every compound term, preceded by a sign, is governed by that sign. Suppose, for example, the quantity a-(6 ed) if we take away the parenthesis, we must -c+d, for as the whole quantity, included by the parenthesis, is to be subtracted, we must change the signs of it. write a not change the signs of the denominator; because every fraction being the expression of a quotient of a division, the sign of the quotient is not changed, if we change, at the same time, the signs, both of the dividend and the divisor, since divided by +, and divided by both give + for the sign of the quotient. - " EXPONENTS. It is very necessary to be well acquainted with the calculations that may be performed with like quantities or like letters, by means of their exponents. Under this head, we shall explain all that belongs to the theory of exponents, and partly repeat what has been already said under the heads of multiplication and division. From this theory we learn, 1st. That like quantities, or like letters, are multiplied the ones by the others, by adding their exponents; thus for example, a3 × a2 — a 3 + 2 — a3; for @xa is in fact the same as aaa Xaa, or a taken five |