be any whole number whatever; hence the method of notation pointed out, (Art. 32), is evident. 87. If a compound quantity is to be diyided by a compound quantity, it frequently occurs, that the division cannot be performed, in which case, the division can be only indicated, in representing the quotient by a fraction, in the manner that has been already described (Art. 8). 38. But if any of the terms of the dividend can be produced by multiplying the divisor by any simple quantity, that simple quantity will be the quotient, of all those terms. Then the remaining terms of the dividend may be divided in the same manner, it they can be produced by multiplying the divisor by any other simple quantity; and by continuing the same method, until the whole dividend is exhausted ; the sum of all those simple quantities will be the quotient of the whole compound quantity. The reason of this is, that as the whole dividend is made up of all its parts, the divisor is contained in the whole as often as it is contained in all its parts. Thus, (ab-cb+ad+cd)--(a+c) is equal to b+d: Forbx(a+c)=ab+cb; and dx(a+c)=ad+cd; but the sum of ab+ch and ad+cd is equal to ab+ cb+ad+cd, which is equal to the dividend; therefrom 6td is the quotient required. Also, (Q2 +3ab+262)+(a+b) is equal to åt-26. For, it is evident in the first place, that the quotient will include the term a, since otherwise we should not obtain a ́. Now, from the multiplication of the divisor a+b by a, arises a2 + ab; which quantity being subtracted from the dividend, leaves a remainder 2ab+262 ; and this remainder must also be divided by a+b, where it is evident that the quolient of this division must contain the term 25 : again, 2b, multiplied by a+b, produces 2ab +2b2; consequently a + 2b is the quotient required; which, multiplied by the divisor a+b, ought to produce the dividend a® + 3ab +2ba. See the operation at length: a+ba? +-3ab+262(a +-25 a2 + ab 2ab + 262 39. ScHolium. If the divisor be not exactly contained in the dividend; that is, if by continuing the operation as above, there be a remainder which canriot be produced by the multiplication of the divisor by any simple quantity whatever; then place this remainder over the divisor, in the form of a fraction, and annex it to the part of the quotient already determined; the result will be the complete quotient. But in those cases where the operation will not terminate without a remainder; it is commonly most convenient to express the quotient, as in (Art 87): 90. Division being the converse of multiplication, it also admits of three cases. CASE I. Ihen the divisor and dividend are both simple quan tities. RULE. 91. Divide, at first, the coefficient of the dividend by that of the divisor; next, to the quotient annex those letters or factors of the dividend that are not found in the divisor ; finally, prefix the proper sign to the result, and it will be the quotient required. Note. Those letters in the dividend, that are common to it with the divisor, are expunged, when they have the same exponent; but when the exponents are not the same, the exponent of the divisor is subtracted from the exponent of the dividend, and the remainder is the exponent of that letter in the quolient. a EXAMPLE 1. Divide 18acby 3ai. -X-X =6X1Xx2-1=61. 18αα2 18 Or, Xa?--? x 32-4-6 X a' Xx=6x: Sec. 3ac 3 Art. 86). 21 Ex. 2. Divide -48a2baca by 16abc." In the first place, 48-:-16=3= the coefficient of the quotient, next, ao bo c2r-abc=a?-- X62-1 Xe?-1= abc; now, annexing abc to 3, we have 3abc, and. prefixing the sign - ; because the signs of the divi dend and divisor are unlike; the result is --3ab which is the quotient required. Or, the operation may be performed thus, -48a2b2c2 48 a 2 62.ca X X--X -3 XaxbXo 16abc a b - 3abc. Ex. 3. Divide - 21x2y3z4 by ----7x*yaz -21'y3z4 21 X13--2 x 49--2 X 24-3=+3xyz, ---7x*yaz =+7 7 Ex. 4. Divide 28a4b5c7 by -70362 cm 16 с 2 28 x a4 c7 23uo88c-7ab2c5= Х 4 x 7 62 +-% x 65-% Xc2-5=-4 Xa? Xb3 Xco=-4a2b3e2. In order that the division, could be effected according to the above rule; it is necessary, in the first place, that the divisor contains no letter that is not to be found in the dividend; in the second place, that the exponent of the letters, in the divisor, do not surpass at all that which they have in the dividend ; finally, that the coefficient of the divisor, divides exactly that of the dividend. -When these conditions do not take place, then, after cancelling the letters, or factors, that are common to the dividend and divisor; the quotient is expressed, in the manner of a fraction, as in (Art. -84). 32 Ex. 5. Divide 48a3b4c2d by 64aob3c'e. The quotient can be only indicated under a fractional form thus, 43a3b5c2d 64a3b3c*e But the coeficients 48 and 64 are both divisible by 16, suppressing this common factor, the coefficient of the numerator will become 3, and that of the denominator 4. The letter a having the same exponent 3 in both terms of the fraction, it follows, that ay is a common factor to the dividend and divisor, and that we can also suppress it. The exponent of the letter b is greater in the divisor than in the dividend; it is necessary to divide bs by 65 13, and the quotient will be bo, or -65-3=h2, which factor will remain in the numerator. With respect to the letter c; the greater power 63 Ans. 4xy. of it is in the denominator; dividing c4 by c?, we have ca c?, or -=44-2=co, therefore, the factor ca will remain in the denominator. Finally, the letters d and e remain in their re: spective places ; because, in the present state, they cannot indicate any factor that is common to either of them. By these different operations, the quotient, in its 362d most simple form, is 4c'e' Note. The division of such quantities belongs.. properly speaking, to the reduction of algebraic fractions. Ex. 6. Divide 36x®y by 9xy. Ans. -cxy Ex. 9. Divide --4axoys by --axy2 Ans. + 4xy. Ex. 10. Divide 16ašb%cx by - 4a3bdy. 4a212CX Ans. dy Ex. 11. Divide --18a3boca by 12a"box. 3c2 2a2bx Ex. 12. Divide 17xyzwa by wyzw. Ans. 1770. Ex. 13. Divide -12a3b%c by -- 6abc. Ans. 2a, boca. Ex. 14. Divide –9x2y2:2 by 2*y*z*. 9 Ans. xaya za 22 Ex. 15, Divide 39a" by 13aR. Ans. 3a4. Ans. 2 2 |