;

dividend. This being divided by a gives for the quotient + b2 multiplying this quotient by the divisor and changing the signs, we have — a b2 + b2; the first term ab3 destroys the first term of the dividend, and the second b3 destroys the other — b3. The mechanical part of the operation will be better understood, if we look for a moment at the multiplication of the quotient a2+ab+b2 by the divisor a b. We see that all the terms reproduced in the process of dividing are those which destroy each other in the result of the multiplication.

45. It sometimes happens that the quantity, with reference to which the arrangement is made, has the same power in several terms both of the dividend and divisor. In this case, the terms should be written in the same column, one under the other, the remaining ones being disposed with reference to another letter. Let there be — a+ b2 + b2 c4.

a2 c4

· a ® + 2 α 1 c2 + bo + 2 b1 c2 + a2 ba,

Arranging the first of these quantities with reference to the letter a, we place in the same column the terms a+ b2 and + 2 ac2, in another, the terms a2 b4 and a2 c4 and in the last column, the three terms + bo, + 2 b1 c2, + b2 c1, disposing them with reference to the letter b, as may be seen in the

next page.

The first term a of the dividend being divided by the first term a of the divisor, gives for the first term of the quotient -a; forming the products of this quotient by all the terms of the divisor, changing the signs of the products in order to subtract them from the dividend, and placing in the same column the terms containing the same power of a, we have, after the reduction of similar terms, the first remainder, which we take for the second dividend.

The first term 2 a b2 of this new dividend, being divided by a2, gives for the second term of the quotient 2 c3 b2; forming the products of this quotient by all the terms of the divisor, changing the signs of the products to indicate their subtraction from the dividend, and placing in the same column the terms containing the same power of a, we have, after the reduction of similar terms, the second remainder, which we take for the third dividend.

The operation being continued in the same manner with the

second remainder and the following ones, we shall have three terms in the quotient. The last being multiplied by all the terms of the divisor, furnishes products which, being subtracted from the fourth remainder, exhaust it entirely. As the division admits of being exactly performed, it follows, that the divisor is a factor of the dividend.

46. The form under which a quantity appears, will sometimes immediately suggest the factors into which it may be decomposed. If we have, for example,

8a6 4a3 b2 + 4 a3 + 2 a3

b3 + 1,

to be divided by 2a3b2+ 1; as the divisor forms the three last terms of the dividend, it is only necessary to see if it is a factor of the three first; but these have obviously for a common factor 4 a3, for 8 a 4a3 b2 + 4a34a3 (2 a3b2+1).

The division is performed at once by suppressing the factor 2 a3 — b2 + 1, equal to the divisor, and the quotient will be 4a3 +1.

After a little practice, methods of this kind will readily occur, by which algebraic operations are abridged.

By frequent exercise in examples of this kind, the resolution of a quantity into its factors is at length easily performed; and it is often rendered very conspicuous, when, instead of performing the operations represented, they are only indicated.

Of Algebraic Fractions.

47. WHEN we apply the rules of algebraic division to quantities, of which the one is not a factor of the other, we perceive the impossibility of performing it, since in the course of the operation we arrive at a remainder, the first term of which is not divisible by that of the divisor. See an example;

1st rem.

α b2 + b3.

a2 + b2

a + b

The first term, a b2, of the second remainder cannot be divided by a2, the first term of the divisor; so that the process is arrested at this point. We can however, as in arithmetic, annex

to the quotient a + b the fraction

mainder for the numerator, and the and the quotient will be

having the re

ab2 + b3 a2 + b2 divisor for the denominator;

b3 a ba

a+b+

a2 + b2

It is evident, that the division must cease, when we come to a remainder, the first term of which does not contain the letter with reference to which the terms are arranged, or to a power inferior to that of the same letter in the first term of the divisor.

48. When the algebraic division of the two quantities cannot be performed, the expression of the quotient remains indicated under the form of a fraction, having the dividend for the nume

rator, and the divisor for the denominator; and to abridge it as much as possible, we should see if the dividend and divisor have not common factors, which may be cancelled (38). But when the terms of the fraction are polynomials, the common factors are not so easily found, as when they are simple quantities. They are in general to be sought by a method analogous to that, which is given in arithmetic for finding the greatest common divisor of two numbers.

We cannot assign the relative magnitudes of algebraic expressions, as we do not give values to the letters which they contain; the denomination of greatest common divisor therefore, applied to these expressions, ought not to be taken altogether in the same sense as in arithmetic.

In algebra, we are to understand by the greatest common divisor of two expressions, that which contains the most factors in all its terms, or which is of the highest degree (27). Its determination rests, as in arithmetic, upon this principle; Every common divisor to two quantities must divide the remainder after their division.

The demonstration given in arithmetic (art. 61) is rendered clearer by employing algebraic symbols. If we represent the common divisor by D, the two quantities proposed might be expressed by the products AD and BD, formed from the common divisor and the factor by which it is multiplied in each of the quantities. This being supposed, if Q stands for the entire quotient, and R for the remainder resulting from the division of AD by BD, we have AD BD × Q + R (Arith. 61); dividing now the two members of the equation by D, we obtain

and since the first member, which in this case must be composed

of the same terms, as the second, is entire, it must follow, that's

R

D

is reduced to an expression without a divisor, that is to say, that R is divisible by D.

According to this principle, we begin, as in arithmetic, by inquiring whether one of the quantities is not itself the divisor of the other; if the division cannot be exactly performed, we divide the first divisor by the remainder, and so on; and that remainder, which will exactly divide the preceding, will be the greatest common divisor of the two quantities proposed. But it will be necessary, in

the divisions indicated, to have regard to what belongs to the nature of algebraic quantities.

We are not, in the first place, to seek a common divisor of two algebraic quantities, except when they have common letters; and we must select from them a letter, with reference to which the proposed expressions are to be arranged, and that is to be taken for the dividend in which this letter has the highest exponent, the other being the divisor.

Let there be the two quantities

3a33a2bab2 — b3,

4a2b-5ab2 + b3,

which are already arranged with reference to the letter a; we take the first for the dividend, and the second for the divisor. A difficulty immediately presents itself, which we do not meet with in numbers, and this is, that the first term of the divisor will not exactly divide the first term of the dividend, on account of the factors 4 and b in the one, which are not in the other. But the letter b being common to all the terms of the divisor and not to those of the dividend, it follows (40) that b is a factor of the divisor, and that it is not of the dividend. Now every divisor common to two quantities, can consist only of factors which are common to the one and to the other; if then there be such a divisor with respect to the two quantities proposed, it is to be looked for among the factors of the quantity 4 a2 — 5 ab + b2, which remains of the quantity 4 a2 b-5 ab2 + b3, after sup pressing b; so that the question reduces itself to finding the greatest common divisor of the two quantities

Заз 3 a2 b + a b2 — b3,
4a25ab+b2.

For the same reason that we may cancel in one of the proposed quantities the factor b which is not in the other, we may likewise introduce into this a new factor, provided it is not a factor of the first. By this step, the greatest common divisor, which can consist only of terms common to both, will not be affected. Availing myself of this principle, I multiply the quantity 3a33a2 bab2 — b3 by 4, which is not a factor of the quantity 4 a 5 ab+b2, in order to render the first term of the one divisible by the first term of the other.

I shall thus have for the dividend, the quantity 12a3 12 a2 b+4ab2

for the divisor the quantity