CASE II. When the Dividend is a Compound Quantity, and the Divisor x Simple one : Divide every term of the dividend by the divisor, as in the former case. EXAMPLES. at bo ab + 62 1. (ab + 62) - 2b, or = a + b. 26 2 10 ab + 15ax 2. (10ab + 15ax) • 5a, or = 2b + 3.7. 5а 30az 482 3. (30az— 482) = 2, or = 300 - 48. 4. Divide 6ab-Sax + a by 2a. 5. Divide 3.x2 – 15 + 6x + 6a by 3r. 6. Divide 6abc + 12abx – 9a’b by 3ab. 7. Divide 10a’r – 15x2 — 25x by 5x. 8. Divide 15d’bc — 15acr” + 5ad by – 5ac, 9. Divide 15a + 3ay, 18y2 by 21a. 10. Divide – 20db2 + 60ab3 by- bab. 1 CASE III. When the Divisor and Dividend are both Compound Quantities: 1. Set them down as in common division of numbers, the divisor before the dividend, with a small curved line between them, and ranging the terms according to the powers of some one of the letters in both, the higher powers before the lower. 2. Divide the first term of the dividend by the first term. of the divisor, as in the first case, and set the result in the quotient. 3. Multiply the whole divisor by the term thus found, and subtract the result from the dividend. 4. To this remainder bring down as many terms of the dividend as are requisite for the next operation, dividing as before, and so on to the end, as in common arithmetic. Note. Note. If the divisor be not exactly contained in the divi'dend, the quantity which remains after the operation is finished, may be placed over the divisor, like a vulgar fraction, and set down at the end of the quotient, as in common arithmetic, 4-6) a3 - 4ac + 4ac - (a*-3ac + c a2- ac - 3a'c + 4ac? a2 -2 4-2) al-6a2 + 12a-8 (-44 + 4 23 - 222 a + x ) 04 – 3.x* ( -d'r + axi-- 43. a ta's 1. Divide a2 + 4ax + 4.42 by a + 2x. Ans. a + 2x. 2. Divide a? - 30% + 3az? --23 by a-%. Ans. a? - 2az + z. 3. Divide 1 by 1 + a. Ans. I-a + ai - a3 + &c, 4. Divide 12x4-192 by 3x – 6. Ans. 4x3 + 8x2 + 16x + 32. 5. Divide a -5a4b + 10a3b2 – 10a-b3 + 5ab4-65 by a? 2ab + b2. Ans. 03 — 3a+b + 3ab? – 63. 6. Divide 4873 – 9.az -64aʼz + 150a3 by 22- 3a. 7. Divide 6 — 384x2 + 3bx4 to by b3 – 36°x + 3bx?- *3. 8. Divide al-x7 by a- *. 9. Divide a + 5a*x + 5ax2 + x3 by a + X. 10. Divide at + 4a 5a - 3264 by a + 2b. 11. Divide 24a4 – 64 by 3a - 2b. ALGEBRAIC FRACTIONS, ALGEBRAIC FRACTIONS have the same names and rules of operation, as numeral fractions in common arithmetic; as appears in the following Rules and Cases. CASE CASE I. To Reduce a Mixed Quantity to an Improper Fraction. MULTIPLY the integer by the denominator of the fraction, and to the product add the numerator, or connect it with its proper sign, + or -- ; then the denominator being set under this sum, will give the improper fraction required. ÉXẢMPLES t and a b 1. Reduce 38, and a to improper fractions. 3 x 5 + 4 15 + 4 19 First, 3 = the Answer. 5 5 5 b a X Ć ai - 6 And, a the Answer ca z? - á? 2. Reduce a + to improper fractions: b az axbta? ab ta First, at the Answer. b the Answer. 3. Reduce 54 to an improper fraction. 38 За 4. Reduce 1- to an improper fraction. Ans. 3axta? $i Reduce 2a to an improper fraction. 44 - 18 6. Reduce 12 + to an improper fraction. 5% 1 = 3a-c 7. Reduce it to an improper fraction. 2.73 30 8. Reduce 4 + 2x to an improper fraction. 5a a Ans. y. CASE II. To Reduce an Improper Fraction to a Whole or Mixed Quantity: DIVIDE the numerator by the denominator, for the integral part; and set the remainder, if any, over the denominator, for the fractional part; the two joined together will be the mixed quantity required. N 2 EXAMPLES EXAMPLES. 16 ab + a 1. To reduce and to mixed quantities. 3 b First, = 16 = 3 = 55, the Answer required. ab + a? a? And, = ab + a = b =a + Answer. b b 2ac – 3a? and 3ax + 4x2 2. To reduce to mixed quantities. a+* 2ac – 3a? 3a? First, - 2ac - 3a:(= 2a Answer. с 2 Ans. 3ax +4.3 x And, = 3ax + 4x2 atm = 3.x + at.c ata 33 3x2 3. Reduce and to mixed quantities. 5 2ax a 2x3 - 2y3 4a ́x 2a? +26 4. Reduce and to whole or mixed quan2a tities. 372 - 3y2 5. Reduce and to whole or mixed x + 9 у quantities. 10a? - 40 + 6 6. Reduce to a mixed quantity. 5a 15a3 + 5a 7. Reduce to a mixed quantity. 3a3 + 2a2 — 20 - 4 CASE III. To Reduce Fractions to a Common Denominator. MULTIPLY every numerator, separately, by all the denominators except its own, for the new numerators; and all the denominators together, for the common denominator. When the denominators have a common divisor, it will be better, instead of multiplying by the whole denominators, to multiply only by those parts which arise from dividing by the common divisor. And observing also the several rules and directions as in Fractions in the Arithmetic. EXAMPLES |