N. B. The learner should constantly bear in mind that the letters, a, b, c, &c. may be used to represent any known number; or they may be used indefinitely, and any number may afterwards be substituted in their place. Again, a b + ab + ab may be written 3 a b, that is, 3 times the product a b; also c times the product a b may be written cab. It may be remarked that a times b is the same as b times a; for a times 1 is a, and a times 6 must be b times as much, that is, b times a. Hence the product of a and b may be written either ab or ba. In the same manner it may be shown that the product cab is the same as a b c. Suppose a = 3, b = 5, and c = =2, then a b c = 3 × 5 × 2, and c a b = 2 × 3 × 5. In fact it has been shown, in Arith. Art. IV., that when a product is to consist of several factors, it is not important in what order those factors are multiplied together. The product of a, b, c, d, e, and f, is written abc def. They may be written in any other order, as a cd bef, or f bedca, but it is generally more convenient to write them in the order they stand in the alphabet. Let it be required to multiply 3ab by 2 cd. The product is 6abcd; ford times 3 a b is 3 ab d, but c d times 3 a b is c times as much, or 3 a b c d, and 2 c d times 3 a b must be twice as much as the latter, that is, 6 a b c d. Hence, the product of any two or more simple quantities must consist of all the letters of each quantity, and the product of the coefficients of the quantities. N. B. Though the product of literal quantities is expressed by writing them together without the sign of multiplication, the same cannot be done with figures, because their value depends upon the place in which they stand. 3ab multiplied by 2 cd, for instance, cannot be written 32 a b c d. If it is required to express the multiplication of the figures as well as of the letters, they must be written 3 ab 2dc, or 3 × 2 abed, or 3.2 ab ed. That is, the figures must either be separated by the letters or by the sign of multiplication. 16. Multiply ax-3ay + xy by 3ay+ax. ax-3ay + xy 3a2 xy-9 a* y2+3 axy3 a2x2 — 9 a2 y2 + 3 ay x2 + a x2y. In adding these two products, the quantity 3 a2 x y occurs twice, with different signs; they therefore destroy each other and do not appear in the result. 17. Multiply 5ad3acd-5a2c by 1 2 ac+2ad. 18. Multiply 13 a2 ry—2 aby2 + 3 c y3 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-2c. 24. Multiply 6ab2ab2 by 2ab-b-1. This operation is sufficiently manifest in the figures. In the letters, I first multiply ab 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. α multiplied by d gives a d-bd; and this subtracted from a c bc gives a c ad+bd. Hence it appears that if two terms having the sign-be multiplied together, the product must have the sign +. .bc 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 3d, If both terms be affected with the sign, the product must have the sign +· Or in more general terms, If both terms have the same sign, whether or, the product must have the sign+, and if they have different signs, the product must have the sign -. 27. Multiply 3a2b−2 ac+5 21 a3b-14 a2bc+35 ab-6 a b c +4 ac2-8 ac-3ab-5. It is generally much easier to trace the effect produced by each of several quantities in forming the result, when the operations are performed upon letters, than when performed upon figures. The following are remarkable instances of this. They ought to be remembered by the learner, as frequent use is made of them in all analytical operations. Let a and b represent any two numbers; a + b will be their sum and a -b their difference. That is, if the sum and the difference of two numbers be multipli ed together, the product will be the difference of the second powers of these two numbers. That is, the product of the sum of two numbers, by itself, or the second power of the sum of two numbers, is equal to the sum of the second powers of the two numbers, added to twice the product of the two numbers. Multiply a-b by a--b. The answer is a2 2ab+b2, which is the same as the last, except the sign before 2 a b. Multiply a2+2 a b + b2 by a + b, that is, find the third power of a + b. Ans. a3+3ab + 3 a b2 + b3. |
