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, X2 23 24 a tx)a ....(1-+ &c. a4 atas X 十 137 And by a process similar to the above, it may be showa 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 1. 2. Let x3 – 3ax2 + 3a? x - a3 be divided by x-. 3. Let a3 +5a? x + 5axo +33 be divided by at... 4. Let 2y3 - 19y2 +267-17 be divided by y-8. 5. Let x5 +1 be divided by x+1, and 36--1 by 6. Let 48x3 — 76ax2 – 64a2 x + 105a3 be divided þy 2x - 3a 7. Let 4x4 - 932 +6x – 3 be divided by 2x2 + 3x – 1. 8. Let x 4 -ax2 +223x -a4 be divided by x2-axtar. 9. Let 6x4 - 96 be divided by 338 - 6, and a5 + x5 by atx. 10. Let 32.05 +243 be divided by 2x+3, and x6 - Q6 by x - a. 11. Let 64 – 3y be divided by b-y, and at +4a2b+ 364 by ū+26. 12. Let x2 +px+q be divided by x+a, and 23 – px+ 9x-rby 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 divisend. 2. Divide this divisor by the remainder, simplified, if necessary, as before ; and so on, for each successive remainsler 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, the terms of the fraction have no conmon 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. (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 o: 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 measure of two of them, and then of that common measure and a third ; and so on to the last D CASE III. To reduce a mixed quantity to an improper fremman. on RULE. Multiply the integral part by the denominator of the fraction, and to the product add tbe numerator, when it is affirmative, or subtract it when negative ; then the result, placed over the denomiuator, will give the improper fraction required. EXAMPLES с b 3X5+2 15+2 17 Ans. 5 5 5 b ахс -6 b And a = Ans. ас — с с 5. Let x x? be reduced to an improper frac20 2x - 7 6. Let 5+ be reduced to an improper frac Зх T be reduced to an improper frac a To reduce an improper fraction to a whole or mixed quantity. 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, join-, ed together, with the proper sign between them, will give the mixed quantity required. EXAMPLES. a2 1 27 axta2 1. Reduce and to mixed quantities. 27 = axta And = (ax+az) --=a+ - Ans. 3 to a whole quantity. ab - 2a2 3. It is required to reduce the fraction to a ab mixed quantity ax 73 |