CASE 1.

To find the greatest cominon measure of the terms of a

fraction.

RULE.

1. Arrange the two quantities according to the order of their powers, and divide that which is of the highest dimensions by the other, having first expunged any factor, that may be contained in all the terms of the divisor, without being common to those of the dividend.

2. Divide this divisor by the remainder, simplified, if necessary, as before ; and so on, for each successive remainser and its preceding divisor, till nothing rerpains, when the divisor last used will be the greatest common measure required ; and if such a divisor cannot be found; tbe terms of the fraction have no coinmon measure.

Note. If any of the divisors, in the course of the operation, become negative, they may have their sigos changed, or be taken affirmatively, without altering the truth of the result ; and if the first term of a divisor should not be exactly contained in the first term of the dividend, the several terms of the latter may be multiplied by any number, or quantity, that will render the division complete. (8)

() in finding the greatest common measure of two quantities, either of them may be multiplied, or divided, by any quantity, which is 'not a divisor of the of her, or that contains no factor which is common to them both, without in any respect changing the result.

It may here, also, be farther added, that the commo , measure, or divisor, of any number of quantities, may be determined in a similar manner to that given above, by first finding the common me..sure of two if them, and then of that common measure and a third ; and so on to the last

D

EXAMPLES.

measure of the

1. Required the greatest common

x 4 - 1 fraction

x5 + x3

•-1)x^+^(

Whence x*+1 is the greatest common measure required.

2. Required the greatest common measure of the frac

23–62 tion x2 +2bx +-62 2+263 +12).3 - 6*x(x

x3 +26x3 +620

3. Required the greatest common measure of the frac

3a2 - 2a-1 cion

4a3 – 20– 3a+1'
3a2 - 2a-1)4a3-2a2 - 3a+1

3

Where, since 2-1)2a’ - 5a +3(2-3, it follows that the

alast divisor a- - 1 is the common measure required.

In which case the common process has been interrupted in the last step, merely to prevent the work overrunning

the page

4. It is required to find the greatest common measure x3-03

of xtar

5. Required the greatest common measure of the fraction

44 --X4
-a? x - ax2 + x3

3

6. Required the greatest common measure of the fraction

24 taxa tas 34 tax3 - a3x – 24

7. Required the greatest common measure of the fraction

7a? – 23ab +662 5a3 - 18a2b+11ab-669

CASE II.

To reduce fractions to their lowest or most simple terms.

RULE.

Divide the terms of the fraction by any number, or quantity, that will divide each of them without leaving a remainder ; or find their greatest common measure, as in the last rule, by which divide both the numerator and denominator, and it will give the fraction required.

EXAMPLES.

a2bc 1. Reduce and

to their lowest terms. 5a2b2

axt x2 abc Here

x2 Ans. And

Ans. 5a2b2 55

axta ata

с

Whence cta is the greatest common measure ;

cx+x2 and c+x)

the fraction required. actas qa

Whence a + b is the greatest common measure ;

and 23-box

x2 - 6x *+6)

the fraction required. x2 +263 +62 x+6 And the same answer would have been found, if x3 212 x had been made the divisor instead of .? +-26x+62.

to its lowest

203 - 16X-6 6. It is required to reduce

3x3-2469 term.

to its lowest terms.