63. The First Formal Rule of Division. The theorems of division are the formal consequences of the fundamental laws of multiplication, 27, namely, and the corresponding laws of addition and subtraction. The rules of division can be deduced in the same way as the rules of subtraction and multiplication (238, 1-5). Equation 1. Because a & x & x bd = The product of two quotients is equal to the quotient of the product of the dividends by the product of the divisors. 64. The Index Law. 1. Required the quotient of a by a3. for by definition of division, p× 1 is equal to p. 2. Required the quotient of a" by a", where m and n are positive integers such that m is greater than n. That is, the exponent of a letter in the quotient is equal to the exponent in the dividend minus the exponent in the divisor. ao. 65. Exponent Zero. According to the index law, the quotient of am by am is am-m, What numerical value has ao? According to rule, 264, = ao . to m factors = 1 1 to m a a a factors = 1. An integer raised to the zero power is unity, i. e., ao = 1. By the rule of signs in 241, 6 and 8, for the products of positive and negative factors, plus times plus or minus times minus produces plus, and plus times minus or minus times plus produces minus. For the present, the analagous laws are assumed for the quotients of positive or negative numbers by positive or negative numbers, divided by +, and divided by, produce +, + divided by and divided by +, produce This law will be established in Chapter X. The quotient of one monomial by another is the product of the quotient of the numerical coefficient by each letter with an exponent equal to its exponent in the dividend minus its exponent in the divisor, and omitting any letter having the same exponent in the dividend and divisor. For 67. Division of Polynomials by Monomials. Equation 2.-Second formal theorem of division. Accordingly the following rule holds: Divide each term of the dividend by the divisor and add the partial quotients. EXAMPLE.-Divide 12 ab3 - 20 a3c — 16 a1be3 by -4a2. 7. 8. 9. 10. 12 a3b2 — 8 a32 + 10 a2¿3 — 14 ab1 by 2 ab2. 36 x3yz2 — 8 x1yz3 — 16x3y2z3 +28.xy3z by —x3yz. 14¿P+1y+1-21xP1y11+49x2Pуa by -7x2p-1y2q−1. 11. Divide 36 m3y+28 m2y2-4 my3 by 4 m2y. 12. Divide 6 ao¿3 — 14 ao¿® + 12 a1xo — a1x2 by — 2a2x3. 13. Divide 23—2n — 3 ¿1y1−2n — 44+mys−n by x2-my2-2n ̧ 14. Divide asyn + x2ny?n + yên by xTMyn. 15. Divide my" — m1−xy1+n+m5-2xyn by m3-xyn−2x ̧ 68. Division of Polynomials by Polynomials. In case both dividend and divisor contain more than one term, the operation of division in Algebra must be performed in the same way as Long Division in Arithmetic. The following rule can be given: Arrange both dividend and divisor according to the powers of some common letter, either both according to ascending or both according to descending powers. Divide the first term of the dividend by the first term of the divisor, and write the result as the first term of the quotient. Multiply the whole divisor by this term, and subtract the product from the dividend, arranging the remainder in the same order of powers as the dividend and divisor. Regard this remainder as a new dividend and repeat the operation till all the terms are brought down. Thus in Long Division in Arithmetic: The reason for the rule is that the whole dividend may be divided into as many parts as may be convenient, and the complete quotient is found by taking the sum of all the partial quotients. [Law X] 3. EXAMPLE 3. Divide xx3. 4x2+5x 3 by x2+2x Arrange the dividend, divisor, and quotient according to descend x y [267, Eq. 2] y x y [268, Ex. 1] |