That is, if the sum and the difference of two numbers be multiplied 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 ab by a-b.

2ab+b2, which is the same as the last,

The answer is a2 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. q3 +3a2b + 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.

Ans. a3-3 ab + 3 a b2 — 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 a b 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.

Examples.

1. How many times is 2 a contained in 6 a b c ? 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 dividend.

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 a3 multiplied by a2 is a2+= a3. In similar cases, division is performed by subtract ing the exponent of the divisor from that of the dividend. a3 divid ed by a2 is a

10. Divide

-2

= = a3.

Ans. 2 abc. b d.

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 + b d + c d.

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

If a d+bd+ c d be divided by a+b+c, the quotient is d. When a compound quantity is to be divided, let the quantity, if possible, be so arranged that the divisor may appear as one of the factors, and then that factor being struck out, the other factor will be the quotient.

19. Divide 12 ab-9 ac by 3 a.

12 ab-9ac3a (4ab3c)

Ans. 4ab3c.

Observe that a is a factor of both terms, and also 3. Hence the quantity 12 a2 b 9 a c, can be resolved into factors; thus 3 (4 ab-3 a c), or a (12 ab-9 c), or 3 a (4 ab-3c). In the last form the divisor 3 a appears as one factor, and the other factor 4 ab-3 c is the quotient.

Note. Any simple quantity, which is a factor of all the terms of any compound quantity, is a factor of the whole quantity; and that factor being taken out of all the terms, the terms as they then stand, taken together, will form the other factor. 20. Divide 8 ab3-16 a3 b c by 2 ab- 4 a2 c. 8 a2 b3 — 16 a3 b2 c4 a b2 (2 a b - 4 a2 c.)

Ans. 4 a b2.

21. Divide 3abc-15 ab2 d+9 a3 bd by 3 a b.

22. Divide 15 a3bc-30 ac2 + 25 a3 c d

23. Divide 36 a13 b2 c— 28 a11 b1 c2 + 40 a3 b® c3

24. Divide 42 a7-84 a10 b c by 1-2 a3b'c.

Algebraic Fractions.

XV. When the dividend does not contain the same letters as the divisor, or but part of those of the divisor, the division cannot be performed in this way. It can then only be expressed. The usual way of expressing division, as has already been explained, is by writing the divisor under the dividend in the form of a fraction. Thus a divided by b is expressed

This gives rise to fractions in the same manner as in arithmetic. It was shown in arithmetic, that a fraction properly expresses a quotient. Algebraic fractions are subject to precisely the same rules as fractions in arithmetic. Many of the operations are more easily performed on algebraic fractions.

In these, as in arithmetic, it must be kept in mind, that the denominator shows into how many parts a unit is divided ; and the numerator shows how many of those parts are used; or the denominator shows into how many parts the numerator is divided.

I shall here briefly recapitulate the rules for the operations on fractions, referring the learner to the Arithmetic for a more full developement of their principles.

a is 24; for 4 of a is, and is 2 times as much. The part of c is; for of c is, and is a times as much.

Hence, to multiply a fraction by a whole number, or a whole number by a fraction, multiply the numerator of the fraction and the whole number together, and divide by the denominator.

Arith. Articles XV. & XVI.