CASE III. To reduce a mixed quantity to an improper fraction. RULE. Multiply the integral part by the denominator of the fraction, and to the product add the numerator, when it is affirmative, or subtract it when negative; then the result, placed over the denominator, will give the improper fraction required. 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, joined together, with the proper sign between them, will give the mixed quantity required. 1. Let 3x+6x2+3αx· = ±a+3b=a+b 10ab-15ax 5a 6x = 26 3x. =5α-8x. 15x be divided by 3x. 2. Let 3abc+12abx-9a2b be divided by 3ab. 3. Let 40a3b3+60a2b2—17ab be divided by -ab. 4. Let 15a2bc-12acx2+5ad2 be divided by --5ac. 5. Let 20ax+15ax2+10x+5a be divided by 5a 1 CASE III. When the divisor and dividend are both compound quantities. RULE. Set them down in the same manner as in division of numbers, ranging the terms of each of them so, that the higher powers of one of the letters may stand before the lower. Then divide the first term of the dividend by the first term of the divisor, and set the result in the quotient, with its proper sign, or simply by itself, if it be affirmative. This being done, multiply the whole divisor by the term thus found; and, having subtracted the result from the dividend, bring down as many terms to the remainder as are requisite for the next operation, which perform as before; and so on, till the work is finished, as in common arithmetic. A 2x2-3αx+a2)4x4-9a2x2+6a3x-a1 (2x2+3ax - a2 4x4-6ax3+2α2x2 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 (ƒ) (ƒ) 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 |