+, the sign may be omitted before this term, but the sign must always be expressed. Great care is requisite in the use of the signs, for an error in the sign makes an error in the result of twice the quantity before which it is written. 3a2bc2-3 c5 a 3 b c2 + 2 c2 + 7 ab +4 b c2 8 c4 But this expression may be reduced. and - 3a5aa8aa7a, 2 b c2 3 b c2 + 4 b c2 - 2 b c2 = 6 b c2 5 b c2= b c2, and -3c2c8 c13 c1—— 11 c1 5 c1—— 6 c1; hence the above quantity becomes 7 a + b c2 + 7 ab-6 c4. To reduce an algebraic expression to the least number of terms, collect together all the similar terms affected with the sign + and also those affected with the sign, and add the coefficients of each separately; take the difference of the two sums and put it into the general result, giving it the sign of the larger quantity. XII. The subtraction of simple quantities, as has already been observed, is performed by giving the sign to the quantity to be subtracted, and writing it before or after the quantity, from which it is to be taken. If it is required to subtract c+d from ab it is plain that the result will be a + b d, for the compound quantity c+d is made up of the simple quantities c and d, which being subtracted separately would give the above result. From 22 subtract 13 7. and 13 7 = 6. 22 6 = 16. - c The result then must be 16. But to perform the operation on the numbers as they stand, first subtract 13, which gives 22 139. This is too small by 7 because the number 13 is larger by 7 than the number to be subtracted, therefore in order to obtain a correct result the 7 must be added; thus 22 13+7=16, as required. First subtract b, which gives a―b. This quantity is too small by c because 6 is larger than b―c Hence to obtain a correct result c must be by the quantity c. 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 ad+ dition. Operation. a2 x + 3by-5 a c3- 16 — 3 a2 x—by + 2 ac2 +22 -2ax+2by-3ac3+6 3b x2-7 a x2 + 13 13bc-3 ax3-8. Ans. 3b x2-13 bc-4 ax3+ 21. 17 ay +13 ay—a—3 42axy- 4 ax 143-17 y XIII. Multiplication of Compound Quantities. Multiplication of compound quantities is sometimes expressed without being performed. To express that a +b is to be multiplied by c-d, it may be written a + bxc-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-d. Multiply a+b by c. It is evident that the whole product must consist of the product of each of the parts by c. a+b 20+4 =24 When some of the terms of the multiplicand have the sign they must retain the same sign in the product. 3 ab x2. 7. 8. Multiply a-b by c, also 23-5 by 4. Since the quantity a-b is smaller than a by the quantity b, the product a c will be too large by the quantity bc. 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. |