CASE I. To find the greatest common 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 remainder and its preceding divisor, till nothing remains, when the divisor last used will be the greatest common measure required; and if such a divisor cannot be found, the terms of the fraction have no common measure. NOTE. If any of the divisors in the course of the ope ration, become negative, they may have their signs 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. (g) (3) 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 other, 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 measure of two of them, and then of that common measure and a third; and so on to the last. D 2. Required the greatest common measure of the fracx3-b2x tion Where a+b is the greatest common measure required. Where, since a—1)2a2 — 5a+3(2a-3, it follows that the last 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 of 4. It is required to find the greatest common measure -a 5. Required the greatest common measure of the fraction a4 --x4 6. Required the greatest common measure of the fraction x2 +α2x2 +αs 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. |