Ex. 6. From 36a-12b+7c Rem. 22a-8b+8 Ans. · In the above example, one row is set under the other, that is, the quantities to be subtracted in the lower line; then, beginning with 14a, and conceiving its sign to be changed, it becomes-14a, which being added to 36a, we have 36a14a 22a; also, -4b, with its sign changed, added to -126, will give 4b-12b=(4—12)b=~8b; in like manner, 7c-7c =0, and 8, with its sign changed, +8. The following examples are performed in the same manner as the last. Ex. 7. = Ex. 8. *+26 Ex. 10. 7x3-3x2-z 6x3-2x2+8x Ex. 12. 7x2-8 9x2+5ab-3x3 3x32x2-5ab-8 67. As quantities in a parentheses, or under a vinculum, are considered as one quantity with respect to other symbols (Art. 10,) the sign prefixed to quantities in a parentheses affects them all; when this sign is negative, the signs of all those quantities must be changed in putting them into the parentheses. Thus, in (Ex. 13), when-cr2 is subtracted from bx2, the result is bxcx2, or —(b−c)x because the sign fixed to (b—c) changes the signs of b and ten +(c-b)x2. prec; or it may be writ Again, in (Ex. 14), when +mx is subtracted from -rx, the result is rx-mx; and, as this means that the sum of re and mx is to be subtracted, that negative sum is to be expressed by (rx+mx)=-(r+m)x. For the same reason, the multinomial quantity -my-ny-aby2ry+6y2, when put into a parentheses, with a negative sign prefixed, becomes -(m—n2+ab+r - 6)y2. Ex. 15. From a-b, subtract a+b. Ans.-26. Ex. 16. From 7xy-5y+3x, subtract 3xy+3y+3x. Ans. 4xy-8y. Ex. 17. What is the difference between 7ax2+5xy-12ay +5bc, and 4ax2+5xy-8ay-4cd. Ans. 3ax-4ay+5bc+4cd. Ex. 18. From 8x2 - 3ax+5, take 5x2+2x+5. Ex. 19. From a+b+c, take —a—b—c. Ans: 3x2-5az. Ans. 2a+26+2c. Fx. 20. From the sum of 3x3 —4ax+3y2, 4y2+5ax—x3, y2—ax+5x3, and 3ax-2x3-y2; take the sum of 5y3 —x2 +x3, ax-x3+4x2, 3x3 — ax- -3y2, and 7y2 — ax+7. Ans. 4x3+4ax-2y3—5x2—7. Ex. 21. From the sum of x2 y2——x2y——3xy2, 9xy2-15— 3x2y2, and 70+2x2 y3-3x2y, subtract the sum of 5x2 y2 —20 +xy', 3x2y-x2y3+ax, and 3xy2-4x2 y2-9+a2x2. Ans. 2ry37x2 yax-a2x2 +84. Ex. 22. From a3x3y3—m3 x3 +3cx-4x2-9: take a2x2 y2 -n2x2+c2x+bx2+3. Ans. (a-a2)x2y2 —(m2 —n2 )x 3 ́ ́+(3c▬▬c2)x−(4+b.) §. III. Multiplication of Algebraic Quantities. x2-12. In the multiplication of algebraic quantities, the following propositions are necessary to be observed. 68. When several quantities are multiplied continually together, the product will be the same, in whatever order they are multiplied. .Thus, axb=b>a=ab. For it is evident, from the nature of multiplication, that the product contains either of the factors as many times as the other contains an unit. Therefore, the product ab contains a as many times as b contains an unit, that is, b times. And the same quantity ab, contains 6 as many times as a contains an unit, that is, a times. Consequently, axb⇒ba= ab; so that, for instance, if the numeral value of a be 12, and of b, 8, the product ab will be 12×8, or 8×12, which, in either case, is 96. In like manner it will appear that abc=cab=bca, &c. 69. If any number of quantities be multiplied continually together, and any other number of quantities be also multiplied continually together, and then those two products be multiplied together; the whole product thence arising will be equal to that arising from the continual multiplication of all the single quantities. Thus, abcd=abcd=abcd. For ab axb, and cd=cXd; if x be put cd, then ab× cd=abxx axbxx; but x is cdcxd, ..ab×x=abc xd=abcd=abcd. = 70. If two quantities be multiplied together, the product will be expressed by the product of their numeral coefficients with the several letters, subjoined. Thus, 7aX5b=35ab. For 7a is =7Xa, and 5b=5×b, :.7a×5b=7×a×5×b =7x5xaxb-35xab=35ab. 71. The powers of the same quantity are multiplied together by adding the indices. 3 Thus, to multiply a2 by a3, it is necessary to write the letter a only once, and to give it for an exponent the sum 2+3, the exponents of the factors; that is, a Xa3a2+3=a5; because a2 = axa, and a3a×a×a; therefore a Xa3=axá xaxaxa=a'. In general, the product of am by a", m and n being always entire positive numbers, is a"+". In fact, am is the abbreviation of a×a×a, &c., continued to m factors, and an is a×a×a, &c., continued to n factors; therefore am Xan = a×a×a×a×a, &c., continued to m+n factors; which (Art. 12) is a n Reciprocally am can be replaced by am Xa". The quantity am is sometimes called an exponential. 72. If two quantities having like signs are multiplied together, the sign of the product will be +; if their signs are unlike, the sign of the product will be. 1. A positive quantity being multiplied by a positive one, the product is positive; thus, +ax+b=+ab, because + a is to be added to itself as often as there are units in b, and consequently the product will be +ab. a 2. A negative quantity being multiplied by a positive one, the product is negative; thus, ax+b=ab; becausé is to be added to itself as often as there are units in b, and therefore the product is ab. Or, since adding a negative quantity is equivalent to subtracting a positive one, the more of such quantities that are added, the greater will the whole diminution be, and the sum of the whole, or the product, must be negative. 1 3. A positive quantity being multiplied by a negative one, the product is negative; thus, +ax-b-ab; because +a is to be subtracted as often as there are units in b, and consequently the product is. —ab. 4. A negative quantity being multiplied by a negative one, the product is positive; thus, ax-bab. For, a×-b ab, that is, when the positive quantity a is multiplied by the negative quantity b, the product indicates that a must be subtracted as often as there are units in b; but when a is negative, its subtraction is equivalent to the addition of an equal positive quantity; therefore, in this case, an equal positive quantity must be added as often as there are units in b. 73. If all the terms of a compound quantity be multiplied separately by a simple one, the sum of all the products, taken together, will be equal to the product of the whole compound quantity by the simple one. For, in the first place, if a+b be multiplied by c, the product will be cacb: Since a+b is to be repeated as many times as there are units in; the product of a by c, that is, ca, is too little by the product of b by c, that is, cb; it is necessary then to augment ca by cb, which will give for the product sought ca+cb, where the term +cb arises from multiplying+bby c. It would be found by reasoning in like manner, that the product of c by a+b must be cacb, where +cb is cx+b. If, in the second place, a-b be multiplied (where a is greater than b) by c, the product will be ca-cb. Since a-b is to be repeated as many times as there are units in c the product of a by c will give too great a result by the pro duct cb; it is necessary then to diminish the product ca by cb, so that the true product is ca-cb. Let, for example, 7-2 be multiplied by 4; the product will be 28-8, or 20; second, or For, 7x4, or 28, is too great by 2x4, or by 8; therefore, the true product will be the first diminished by the 288, that is 20. In fact, 7-2, or 5×4=20. -cb of the product, is the product of —b by c. The term It would be found, by reasoning in like manner, that the product of c by a-b, must be ac-bc, the same as in the preceding, and in which the term —bc is the product of c by b. If, in the third place, a+b+d be multiplied by c, the product will be ca+cb+cd. For, let a+b be designated by e; then, e+d multiplied by c is equal to ce+cd; but ce is equal to c × (a+b) = ca + cb, because e is equal to a+b; therefore (a+b+d) ×c=ca+cb +cd. Also, if (a+b)-d be multiplied by c, the product will be ca+cb-cd; for let (a+b)=e, then (e-d)Xc=ce-cd= c(a+b)-cd-ca+cb-cd. Finally, it may be demonstrated in like manner, that if any polynomial, a+b-d+e-f, &c., be multiplied by c, the product will be ca+cb-cd+ce-cf, &c. Also, if a quantity c be multiplied by any polynomial a+b−d+e, &c., the product will be ac+bc-detec, &c. 75. If a compound quantity be multiplied by a compound quantity, the product will be equal to every term of one factor, multiplied by every term of the other factor, and the products added together. Let, in the first place, a+b be multiplied by c+d: a+b taken c times is ca+cb, as we have already proved; but this product is too little by the binomial a+b repeated d times, it is necessary then to add to it da+db, and we will have ca+cb +da+db for the product sought; in which the term +db arises from the multiplication of +b by +d. -d, Suppose, in the second place, that a+bis multiplied by cthe product will be ca+cb-da-db. Because the product of a+b by c, that is, ca+cb, is too great by that of a+b by d, which is da+db; we will have therefore the true product equal to ca+cb-da--db, where the term-db is the product of +b by -d; in multiplying c-d by a+b, we will find that bd is the product of -d by +b. Let, in the third place, a-b be multiplied by c-d; the product will be ca-cb--da+db. For, the product of a-b by c, that is, ca-cb, is too little by |