24. Multiply 6a2b+2ab2 by 2 ab-b— 1.

171-45-76 +20= 191 — 121 = 70.

26. Multiply ɑ—b

by

c-d.

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-be 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 a d-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 +.

ac

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

Product

21 a3b2-14 a' b c + 35 a b — 6 a3b c + 4 a3c2—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.

That is, if the sum and the difference of two numbers be multipli ed together, the product will be the difference of the second

of these two numbers.

powers

a+b

a+b

a2 + ab

a b + b3

a2+2ab+b2.

That is, the product of the sum of two numbers, vy 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 ab by a-b.

The answer is a2. 2ab+b2, which is the same as the last, except the sign before 2 a b.

Multiply a +2 a b + b2 by a + b, that is, find the third power of a+b.

Ans. a3ab + 3 a b2 + b3.

:

This is expressed in words thus the third power of the first, plus three times the second power of the first into the second, plus three times the first into the second power of the second, plus the third power of the second.

Multiply a-2ab+b by a-b.

Ans. a3 — 3 a3 b + 3 a b3 —b3. Which is the same as the last, except the signs before the second and last terms.

Instances of the use of the above formulas will frequently occur in this treatise.

Division of Algebraic Quantities.

XIV. The division of algebraic quantities will be easily performed, if we bear in mind that it is the reverse of multiplication, and that the divisor and quotient multiplied together must reproduce the dividend.

The quotient of a b divided by a is b, for a and b multiplied together produce a b. So ab divided by b gives a for a quotient, for the same reason.

If 6 a b c be divided by 2 a, the quotient is 3 b c.

For in all these instances the quotient multiplied by the divisor, produces the dividend 6 abc.

1. How many

Examples.

times is 2 a contained in 6 abc?

Ans. 3b c times, because 3 b c times 2 a is 6 a b c.

2. If 6 a b c be divided into 2 a parts, what is one of the parts?

Ans. 3bc; because 2 a times 3 b c is 6 a b c.

Hence we derive the following RULE. Divide the coefficient of the dividend by the coefficient of the divisor, and strike out the letters of the divisor from the divi

dend.

Observe that 4 a3 is the same as 4 a a a and a2 is the same as a a; 4 a a a divided by a a gives 4 a for the quotient.

It was observed in multiplication, that when the same letter enters into both multiplier and multiplicand, the multiplication is performed by adding the exponents, thus a multiplied by a* is a+= a. In similar cases, division is performed by subtracting the exponent of the divisor from that of the dividend. a' divided by a2 is a

10. Divide

-2

= a3.

The division of some compound quantities is as easy as that

of simple quantities.

If a+b+c be multiplied by d the product is

d(a+b+c) or ad+bd+c d.

Therefore if a d+bd+cd be divided by d, the quotient is a+b+c.