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 + 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.

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 ac 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.

16. Multiply a x-3ay + xy by 3ay+ax.

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.

If 3b+2 c be multiplied by 2 a only, the product will be too large by 3 b times 36+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 +36c+2 by 4ab-2 c.

24. Multiply 6 ab+2ab by 2ab-b- 1.

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. α -b multiplied by d gives ad-bd; and this subtracted from - b 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 +.

a c

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—2ac+5

21 a3b2 — 14 a2bc + 35 a b — 6 a3b c + 4 a2c2-8ac-3a2b—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.