evident that the co-efficients of these polynomials have not a common factor.

70. These examples are sufficient to point out the course the beginner is to pursue, in finding the greatest common divisor of two polynomials, which may be expressed by the following general

RULE.

I. Take the first polynomial and suppress all the monomial factors common to each of its terms. Do the same with the second polynomial, and if the factors so suppressed have a common divisor, set it aside as forming a part of the common divisor sought.

II. Having done this, prepare the dividend in a such a manner that its first term shall be divisible by the first term of the divisor; then perform the division, which gives a remainder of a degree less than that of the divisor, in which suppress all the factors that are common to the co-efficients of the different powers of the principal letter. Then take this remainder as a divisor, and the second polynomial as a dividend, and continue the operation with these polynomials, in the same manner as with the preceding.

III. Continue this series of operations until a remainder is obtained which will exactly divide the preceding remainder; this last remainder will be the greatest common divisor; but if a remainder is obtained which is independent of the principal letter, and which will not divide the co-efficients of each of the proposed polynomials, it shows that the proposed polynomials are prime with respect to each other, or that they have not a common factor.

EXAMPLES.

1. Find the greatest common divisor between the two poly. nomials.

Hence, 2a+3b+c is the greatest common divisor.

After arranging the two polynomials, the division may be performed without any preparation, and the first remainder will be,

To continue the operation, it is necessary to take the second polynomial for a dividend, and this remainder for a divisor, and multiply the new dividend by 66-10c, or simply 36-5c, since 2 is a factor of the first term of the dividend. But we are not at liberty to multiply by 3b-5c, if it is a factor of the remainder. Therefore, before effecting the multiplication, we must see if 36-5c will exactly divide the first remainder; we find that it does, and gives for a quotient 2a+3b+c: whence it follows that the remainder can be put under the form

(3b-5c) (2a+3b+c).

Now, 36-5c is a factor of this remainder, and is not a factor of the new dividend. For, being independent of the letter a, if it was a factor of the dividend it would necessarily divide the co-efficient of this letter in each of the terms, which it does not; we may therefore suppress it without affecting the greatest common divisor.

This suppression is indispensable, for otherwise a new factor would be introduced into the dividend, and then the two polynomials containing a factor they had not before, the greatest common divisor would be changed; it would be combined with the factor 36-5c, which should not form a part of it.

Suppressing this factor, and effecting the new division, we obtain an exact quotient; hence

2a+3b+c is the greatest common divisor.

REMARK. The rule for the greatest common divisor of two polynomials, may readily be extended to three or more polynomials. For, having the polynomials A, B, C, D, &c. if we find the greatest common divisor of A and B, and then the greatest common divisor of this result and C, the divisor so obtained will evidently be the greatest common divisor of A, B and C ; and the same process may be applied to the remaining polynomials.

2. Find the greatest common divisor of x*—1 and x2+x3.

Ans. 1+x2.

3. Find the greatest common divisor of 4a3-2a2-3a+1 and 3a2-2a-1. Ans. a-1.

4. Find the greatest common divisor of a*-x1 and a3—a2x— ax2 +x3. Ans. a3-x3.

5. Find the greatest common divisor of 36a-18ao — 27a*+9a3 and 27ab2-18a1b3-9a3b3.

6. Find the greatest common divisor of

Ans. 9a (a-1).

qnp3 +3np3q3 — 2npq3 — 2nq* and 2mp3q3—4mpʻ—mp3q+3mpq3.

Ans. p-q.

7. Find the greatest common divisor of the two polynomials 15a+10a b+4a3b2+6a2b3-3ab*

12ab+38a2b+16ab-10b".

CASE II.

Ans. 3a+2ab-b2.

71. To reduce a mixed quantity to the form of a fraction.

RULE.

Multiply the entire part by the denominator of the fraction: then connect this product with the terms of the numerator by the rules for addition, and under the result place the given denominator.

CASE III.

72. To reduce a fraction to an entire or mixed quantity.

RULE.

Divide the numerator by the denominator for the entire part, and place the remainder, if any, over the denominator for the fractional part.

73. To reduce fractions having different denominators to equiva

lent fractions having a common denominator.