Examples. 2. Divide a +5a*x+10a3r© +- 10a2x2+5ax*+x3. Ans. a + 3 a® x + 3 ax2 + r. 3. Divide 12 x3- 13 x 34 x + 35 x by 4x-7x. — xo 5. Divide y 3y'x2 + 3y2x2 + x by y3-3yx + 3yx2 — 23. Ans. y + 3yx + 3yx2 + x3 10a3x3 + 5ax1 — x3 by a' Ans, as ·3ax +3ax1 Miscellaneous Examples. r. 1. Divide a3-r3 by a — x. Ans, a + a'r + a2r2 + ax3 + x*. 6. Divide r3 + 3x3y + 3xy2 + y3 by x2 + 2xy + y2. Divide 6x96 by 3r - 6. Ans. 2x3 + 4x2 + 8x + 16. 8. Algebraic Fractions are subject to the same rules as are the fractions of common arithmetic, and their reduction requires only the application of the principles laid down in the preceding rules. PROBLEM I. To reduce a mixed quantity to an improper fraction. Rule. Multiply the integer by the denominator of the fraction, and to the product annex the numerator with its proper sign; under this same place the former denominator, and the result will give the improper fraction required. Here the integral part × denom. of the fraction + the numerato? = 3a x 5a2 + 2x = 15a3 + 2x. To reduce an improper fraction to a whole or mixed quantity. Rule. Observe which terms of the numerator are divisible by the denominator without a remainder, the quotient will give the integral part; to this annex the remaining terms of the numerator, with the denominator under them, with their proper signs, and the result will be the mixed quantity required. Example 1. to a whole or mixed quantity. Reduce a2 + ab + b2 a To reduce fractions to a common denominator. Rule. Multiply each numerator into every denominator but its own for the new numerators, and multiply all the denominators for a common denominator. 5x multiplied by 7 = 35x ; and 2x-3 subtracted from 35% leaves 33x+3 To reduce a fraction to its lowest terms. Rule. Observe what quantity will divide all the terms of both numerator and denominator without any remainder: divide them by this quantity, and the fraction is reduced to its lowest terms. 1. Reduce Examples. 14237ax + 2x2 35 x to a common denominator. Note. The co-efficient of every term of both numerator and denominator is divisible by 7, and a enters also into every term; +7x will therefore divide both numerator and denominator without a remainder. For 14x+7ax + 21.x2 = 2x2 + a + 3x ; =5x. Hence, the frac. in its lowest terms, is 2a2 + a +3 7x 5x a2 + ab + b2 as Scholium. When, therefore, we have occasion to reduce algebraic fractions to their lowest terms, by finding the greatest common measure of the numerator and the denominator, as in common arithmetic, it is evident that the rule which we have given for the solution of the foregoing problem answers every practi cal purpose, since it is only requisite to discover what quantity will divide both numerator and denominator, and that the quantity must therefore be the greatest common measure. PROBLEM V. (43.) To add fractional quantities. Rule. Reduce the fractions to a common denominator, as in Problem III. Add all the numerators together, and below their sum set down the common denominator, and it will give the sum of the fractions required. Examples. |