From a subtract b -C. First subtract b, which gives a This quantity is too small by c because 6 is larger than b-c by the quantity c. Hence to obtain a correct result c must be added, thus ab+c. This reasoning will apply to all cases, for the terms affected with the sign in the quantity to be subtracted diminish that quantity; hence if all the terms affected with + be subtracted, the result will be too small by the quantities affected with these quantities must therefore be added. The reductions may be made in the result, in the same manner as in addition. Hence the general RULE. Change all the signs in the number to be subtracted, the + signs to, and the signs to +, and then proceed as in addition. a2x+3by-5 a c3 — 16 -2 a2x+2by-3ac3 +6 3b x2-7 a x2+13 13bc-3 ax3-8. Ans. 3b x-13bc-4ax+21. 17 a2y+13 ay—a—3 42axy-4 ax 17 ax-2axy-5 143-17 y 6. From Subtract 7. From a+3abc-1 1+3abc-a. 3abz+2ab-7 z Subtract 2 ab-7z-2 abz. Multiplication of Compound Quantities. XIII. Multiplication of compound quantities is sometimes expressed without being performed. To express that a +b is to be multiplied by cd, it may be written a+bx c — d with a vinculum over each quantity, and the sign of multiplication between them; or they may be each enclosed in a parenthesis and written together, with or without the sign of multiplication; thus (a + b) × (c—d) or (a + b) (c—d). In the expression a+b (c-d), b only is to be multiplied by c Multiply a+bby c. d. It is evident that the whole product must consist of the product of each of the parts by c. 6. Multiply Ans. 3 abef+2cdef. 5ac+be+3 cd by 2 e. Ans. 10 ace+2bce+6cde. 6 a2b+b2 c3 by 3 a b3. b c2 d3 +52 a2 b2 + 13 b3 c2 d 2 abd +3ab x2 + a b x2. 3 abx. ax2+3 ab x2 by 13 a b3 3·· When some of the terms of the multiplicand have the sign they must retain the same sign in the product. Since the quantity a-b is smaller than a by the quantity b, the product ac will be too large by the quantity b c. This quantity must therefore be subtracted from a c. When both multiplicand and multiplier consist of several terms, each term of the multiplicand must be multiplied by each term of the multiplier. It is evident that if a +3 be taken c times and then d times, and the products added together, the result will be c+d times a+b. 16. Multiply ax-3ay + xy by 3ay+ax. In adding these two products, the quantity 3 a2xy occurs twice, with different signs; they therefore destroy each other and do not appear in the result. 17. Multiply 5ad3acd-5 a c by 2 ac+2ad. 18. Multiply 13 a2 ry-2 a b y2 + 3 cy3 If 3b+2 c be multiplied by 2 a only, the product will be too large by 3 b times 3b+2c; hence this quantity must be multiplied by 3 b, and the product subtracted from 6 a b + 4 a c. This result may be proved by multiplying the multiplier by the multiplicand, for the product must be the same in both cases. 23. Multiply 2ad + 3bc+2 by 4ab-2 c. 24. Multiply 6 ab+2ab by 2ab-b2-1. 171 — 45 — 76 + 20 = 191 — 121 = 70. This operation is sufficiently manifest in the figures. In the letters, I first multiply a -b by c, which gives a c -bc; but the multiplier is not so large as c by the quantity d, therefore the product a c-bc is too large by d times a-b; this then must be multiplied by d and the product subtracted. a -b multiplied by d gives ad-bd; and this subtracted from ac-bc gives a c— -bc- ad+bd. Hence it appears that if two terms having the sign-be multiplied together, the product must have the sign +. From the preceding examples and observations, we derive the following general rule for multiplying compound quantities. 1. Multiply all the terms of the multiplicand by each term of the multiplier, observing the same rules for the coefficients and letters as in simple quantities. 2. With respect to the signs observe, 1st, That if both the terms which are multiplied together, have the sign+, the sign of the product must be +. 2d, If one term be affected with +, and the other with, the product must have the sign |