78. To determine the law of the SIGNS of the quotient. Since the quotient multiplied by the divisor produces the dividend, it follows, + divided by +, and divided by divided by +, and + divided by Thus, ab÷+b= +a, +ab÷ b== produce +, abb=+a, — ab÷+b=a. - Hence, in division, as in multiplication, Like signs produce +, and unlike signs produce CASE I. 79. When the divisor is a monomial. From the preceding principles is derived the following RULE. Divide the coefficient of the dividend by that of the divisor; and to the result annex the letters of the dividend and divisor, each with an exponent equal to that in the dividend diminished by the exponent in the divisor, omitting all letters whose exponents become zero. Make the terms in the quotient positive when the dividend and divisor have like signs, and negative when they have different signs. When the dividend has more than one term, divide each separately, and connect the results by their proper signs. 7. Ans. axy. Divide 8abc + 16 a b c — 4 a2 c2 by 4 a2c. 8. Divide 9abc-3a2b+18 abe by 3 ab. 12. Divide 6 (a + b)3x2y by 3 (a+b)2xy. Ans. a2m-n ̧ Ans. 2 (a + b) x. 13. Divide 3 a3 (a - b) + 9a (a+b) by 3 a. Ans. a2 (ab) + 3 (a + b). 16. Divide 15,(x + y)2 — 5 a (x + y) + 10 b (x + y) 17. Ans. 3 (x + y) + a2b. — Divide 12a-8a2b+16 a3 x 10a-2y by 2 a2. α + a x2 + 2 + a x2+m by x”. Ans. a+ax-2 1 + a x2 + a xm. CASE II. 80. When the divisor is a polynomial. 1. Divide a3 + 5 a2 x + 5 a x2 + x3 by a + x. The divisor and dividend are both arranged with reference to the decreasing powers of a. Now, since the first term of the dividend, as arranged, must equal the product of the first term of the divisor by the first term of the quotient (Art. 73), we have a2 for the first term of the quotient. The product of the divisor by this term, subtracted from the dividend, gives a new dividend. With this dividend, arranged as the original one, we proceed in like manner as with that, and obtain 4ax, the next term of the quotient; and so on. The divisor is written on the right of the dividend, for the convenience of multiplying it by the several terms of the quotient, as they are found. It may be written on the left. Hence, the RULE. Arrange both dividend and divisor according to the decreasing powers of some letter. Divide the first term of the dividend by the first term of the divisor, and write the result for the first term of the quotient. Multiply the whole divisor by this term, and subtract the product from the dividend. Regard the remainder as a new dividend, arrange it and find the next term of the quotient, in the same manner as beforeand so proceed until all the terms of the dividend are exhausted. When it happens that the division cannot be exactly performed, the quantity remaining undivided may be written with the divisor under it, in the form of a fraction, and annexed to the quotient. In general, division terminates when the first term of the arranged remainder will not contain the first term of the divisor. EXAMPLES. a2 + 4 a2 b2 + 16 b1 a2 2 a b + 4 b2 4x2-7x) 12x-13x-14x+8x2+1(3x2+2x2+2+ 14x+1 4x2-7x 14x+1, Remainder. Here, the divisor is written on the left of the dividend. b2 4. Divide 8a-4a2b-6 ab23b by 2 a-b. Ans. 4a2-3 b2. 5. Divide 3+3 a b2 — 4 a2 b — 4a3 by a + b. 7. Divide 21 a3 — 2165 by 7a7b. Ans. 3a3 a3 b + 3 a2 b2 + 3 a b3 + 3 b1. 8. Divide xy + 2 y2 22-24 by x2 + y2 — z2. +1 12. Divide "+1 + x2 y + xy" + y2+1 by x2+y". Ans. xy. 13. Divide 29x26xy-y2 by x2+3x+y. 14. Divide 1+ a by 1 α. Ans. x2 3 x y. Ans. 12a + 2 a2 + 2 a3 +, etc. In examples of this kind the division does not terminate, there being a remainder, however far the operation may be carried. 17. Divide a + a® b2 + a+b+ + a2 b® + b3 by a1 + a3 b + a2 b2 + a b3 + b2: Ans. atab + a2 b2 — a b3 + b2. DIVISION BY DETACHED COEFFICIENTS. 81. Division, as well as multiplication (Art. 74), sometimes may be conveniently performed by operating on the |