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Note, 1... If the divisor be not exactly contained in the dividend, the quantity that remains after the division is
finished, must be placed over the divisor, at the end of the quotient, in the form of a fraction : thus (f)
(f) In the case here given, the operation of division may be considered as terminated, when the highest power of the letter, in the first, or leading term of the remainder, by which the process is regulated, is less than the power of the first term of the divisor ; or when the first term of the divisor is not contained in the first term of the remainder; as the succeeding part of the quotient, after this, instead of being integral, as it ought to be, would necessarily become fractional,
* The division of quantities may also be sometimes carried on, ad infinitum, like a decimal fraction; in which case, a few of the leading terms of the quotient will generally be sufficient to indicate the rest, without its being necessary to continue the operation ; thus,
Where the law, by which either of these series may be continued at pleasure, is obvious.
AlgeBRAIC fractions have the same names and rules of operation as numeral fractions in common arithmetic ; and the metheds of reducing them, in either of these branches, to their most convenient forms, are as follows :
And by a process similar to the above, it may be shown
To find the greatest common measure of the terms of a
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 remain "er and its preceding divisor, till nothing reinains, 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 operation, 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 com
(*) 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