sign × between them, and a vinculum over the compound quantity. II. Multiply each of the terms of the compound quantity by the simple one, by the foregoing rules; and place the several products, with their proper signs, in a line; these will form the product required o. 3 ax -- OPERATION. 262 +z Зах Explanation. 3 ax Xab=3a2bx... - 26? x ab =- - 2 ab3 and ≈ Xab = abz; these products placed in or - 2 ab3 + abz product. der, with their proper signs, give the product required. Art. 43. Or thus, · 2 b2 + z × ab = 3 a2 bx − 2 a b3 + abx, product. • This rule follows from the preceding; for the product of the whole multiplicand into the multiplier, is evidently equal to the sum of the products of the several parts of the former, multiplied into the latter; the truth of which is likewise demonstrated geometrically, in the first proposition of the second book of Euclid's Elements. 23. Multiply 2 ay — ay2+3 ay3 into 4by. Prod. 8 aby2 —4 aby3 +12 aby1. 24. Multiply-a2+2x-6 by xy. 25. Multiply 2—2 xz+22 into -4. 26. Multiply 3-abx+2ab into 12. Prod. -a2xy + 2x2y—6xy. Prod. -4x2+8xz-4z2. Prod. 36-12 abx+24 ab. 45. When both factors are compound quantities. RULE I. Place the multiplier under the multiplicand, and let them both stand in the same order. II. Multiply the first or left hand term of the multiplier, into every term of the multiplicand, and place the products, with their proper signs, in a line below, by the preceding rule. III. Multiply the second term of the multiplier into every term of the multiplicand; and place the products, with their proper signs, under the former products, observing to set like quantities (when they occur) under each other. IV. Proceed in this manner with every term of the multiplier, then add all the products together, and the sum will be the product required P. 27. Multiply 2a+3b into 4 a+5b. OPERATION. 2a+3b 8 a2 + 12 ab +10 ab+ 15 b2 Prod. 8a2+22 ab + 15 b2 Explanation. I first multiply the upper line by 4 a, and the product 8 a2+ 12ab is the first line of the product. I then multiply the top line by 5b, and the product 10 ab + 15 62 is the second line of the product; likewise I place the ab's under each other. I then add these two products together, and the sum is the last line, or product required. P Here the whole multiplicand is considered as multiplied into the whole multiplier, when every member of the one is multiplied into every member of the other, and their products collected together, (as far as is practicable,) into one sum the truth of this follows from what has been said in the note on the preceding rule, and it may be illustrated and confirmed by particular examples in numbers, such as the example given at the end of the note on Art. 42, Prod. 3x2+2 xy-8 y2. . . .=x+2yx ..=x+2yx 3x-4y 36. Multiply a+x into a+r. Prod. a2+2 ax+x2. 37. Multiply 2x+3y into 3x-4y. 38. Multiply x+1 into x-1. 39. Multiply 3+2 into 4—2. Prod. 6 x2+xy-- 12y2, Prod. x2-1. Prod. 12+-2o. 40. Multiply 2 a2-3 a-4 into 3 a3-5 a. 22a3 +15 a2+20 a. Prod. 6a-9 a1 41. Multiply 2x2+3 xy-5 y2 into x-y. Prod. 2x+3y -7 x2y2-3 xy3+5 y1. * 42. Multiply 4x-5 a-2b into 3x-2a+5b. Prod. 121o23 ax+14 bx+ 10 a2-21 ab— 10 bo. DIVISION. To divide one simple quantity by another. 46. When both terms consist of the same letters, and the dividend contains the divisor some number of times exactly. RULE I. Place the dividend above the divisor, with their proper signs, and a small line between them, like a fraction. II. If the terms have like signs, make the sign of the quotient; but if they have unlike signs, make that of the quo tient III. Divide the coefficient of the upper term by that of the lower, and the result will be the coefficient of the quotient a. 4 The rule for the coefficients is the same as that for simple division in Arithmetic. (Art. 37. Part I.) With respect to the subtraction of the index of the divisor from that of the dividend, in order to find the index of the quotient, it may be observed, that division being the converse of multiplication, the methods of operation in one will evidently be the converse of those in the other. Wherefore, since it has been shewn, (Art. 42. and note,) that addition of the indices of like letters produces multiplication, it follows, that subtracting the indices (viz. that of the divisor from that of the dividend) will produce division; or in other words, if the index of the divisor be subtracted from that of the dividend, the remainder will be the index of the quotient. If one quantity be divided by another without remainder, it is plain that the quotient will be such, that being multiplied by the divisor, the resulting product will equal the dividend; now it is equally plain that the sign of the quotient must be such, that when the quotient is multiplied by the divisor, the product will (not only be equal to, but) have the same sign with the dividend, according to the rule for the signs in Art. 42. Wherefore it follows, that 1. When both terms (namely the divisor and dividend) are +, the quotient must likewise be + ; for + in the divisor must have+ in the quotient, to produce in the dividend. 2. When both terms are, the quotient must be +; for-in the divisor must have in the quotient, to produce in the dividend. and the other 3. When either of the terms is +, the quotient must be; for in the divisor must have in the quotient, to produce — in the dividend. in the quotient, to produce + in the Bb IV. Subtract the index of each letter in the lower term, from the index of the same letter in the upper, and place the remainder as an index over that letter in the quotient. V. Having proceeded in this manner with all the letters concerned, the result, connected with the sign and coefficient (found as above) will be the quotient required. If any letter has the same index in both terms, that letter is cancelled from the operation, and does not come into the quotient. EXAMPLES. 1. Divide 12 ax3 by 4a2xa. OPERATION. 12 a6x8 4 a3x4 Explanation. Having placed the terms, I find that they -=3 a3x quot. have like signs; wherefore that of the quotient is + understood: next I divide 12 by 4, and the quotient 3 is the coefficient of the quotient: then I subtract 3, the index of a in the lower term, from 6, the index of a in the upper, and the remainder 3 is the index of a in the quotient. I do the same by x, subtracting 4 from 8, and the remainder 4 is in like manner the index of x in the quotient. 2. Divide 56x+y6 by -7x2y2. OPERATION. 56x4y6 =-8x2y1 quot. -7x2yz Explanation. Unlike signs here give, and 56÷7 =8 the coefficient; also 4-2-2 the index of a, and 6-2=4 the index of y. 3. Divide 28 a 24 by —7 ax2, and — Sr3 by 2x. 2 y5z4 = y223. =yz2. - xyz The following is a summary of the whole doctrine brought into one point of Wherefore in division (as in multiplication) like signs always produce +, and unlike |