2. 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, And by a process similar to the above, it may be shown that Where the law, by which either of these series may be continued at pleasure, is obvious. EXAMPLES FOR PRACTICE. 1. Let a2-2ax+x2 be divided by a-x. 2. Let x3-3ax2+3a2x - α3 be divided by x-a. 3. Let a3+5a2 x + 5ax2 + x3 be divided by a+x. 4. Let 2y3-19y2 + 26y-17 be divided by y-8. 5. Let x5+1 be divided by x+1, and x6-1 by x-1. 6. Let 48x3-76ax2-64a2x+105a3 be divided by 2x-3a 7. Let 4x4-9x2+6x-3 be divided by 2x2+3x-1. 8. Let x 4-a2x2+2a3x-a4 be divided by x2-ax+a2. 9. Let 6x4-96 be divided by 3x-6, and a5 + x5 by α+x. 10. Let 32x5+243 be divided by 2x+3, and x-a6 by x-a. 11. Let b4-3y be divided by b-y, and a4+4a2b+ 364 by a+26. 12. Let x2+px+q be divided by x+a, and x3-px2+ qx-rbyx-. OF ALGEBRAIC FRACTIONS. ALGEBRAIC fractions have the same names and rules of operation as numeral fractions in common arithmetic; and the methods of reducing them, in either of these branches, to their most convenient forms, are as follows: 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 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 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 RULE. Divide the numerator by the denominator, for the integral part, and place the remainder, if any, over the denominator, for the fractional part; then the two, joined together, with the proper sign between them, will give the mixed quantity required. It is required to reduce the fraction ax-23 to a antity to reduce the fraction ab-2a2 to a |