- 5. Let a+1 be divided by x+1, and x-1 by x-1. 6. Let 48x-76ax2 - 64a2x + 105a3 be divided by 2x-3a. 7. Let 4x*-9x2+6x-3 be divided by 2x2+ 3x-1. 8. Let x*-a2x2+2ax-a be divided by x2ax + a2. 9. Let 6x-96 be divided by 3x-6, and ao - x by a-x. 10. Let b*-3y be divided by b−y, and a*+ 4a2b+3b1 by a + 2b. 11. Let x2 + ax + b be divided by x+p, and x3px2+qx-r by x-a. OF ALGEBRAIC FRACTIONS. (F) 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: To find the greatest common measure of the terms of a fraction. RULE. 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; then divide this divisor by the remainder, simplified, if necessary, as befare; 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. When 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 (m). EXAMPLES. 1. Required the greatest common measure of the fraction x4 - 1 Where 2+1 is the common measure required. (m) 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 common measure, or divisor, of any number of quantities may be determined in a Where x+b is the common measure required. 3. Required the greatest common measure of 3a2-2a-1 4a3-2a2-3a + 1° 3a2-2α-1)4a3 — 2a2 — 3a + 1 Where a-1 is the common measure required. 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. 10. Required the greatest common measure of 7 a2-23ab + 6b2 5a3-18a2b+ 11ab2 - 6b2 CASE II. To reduce fractions to their lowest or most simple terms. RULE. Divide each of the terms of the fraction by any number, or quantity, that will divide 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. |