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 2 x4 a + x) a- 3x* (a3 - a'r + ax? - 33 ata @* + aux 2az + . 1. Divide a® + 4ax + 4x? by a +- 23. Ans. at 23. 2. Divide a? 3u>z +- Jaz? z3 by a – 2. Ansa? 3. Divide 1 by 1 ta. Ans. I-a + a2 a3 + &c. 4. Divide 12x 192 by 323 6. Ans. 4.33 + 8x2 + 16.3 + 32. 5. Divide as 5a46 + 10a362 – 10a2b3 + 5a64 65 by ao – 2ab + 62. Ans. a3 Ja?b + 3ab2 - 63. 6. Divide 487% 96az? – 64a2z + 150a3 by 2z Sa. 7. Divide 66 -36*x+36°x*- x6 by 63-362x+36x-23, 3. Divide a? x07 by a - X. 9. Divide a3 + 5a2x + 5ax2 + x3 by a + x. 10. Divide a4 + 4a2b2 -3264 by a + 26. 11. Divide 2494 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 Imfiroper Praction. 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. EXAMPLES. 6 1. Reduce 3ş, and a - to improper fractions. and a a a2 a2 2. Reduce at to improper fractions. b a axb + aa ab + a2 First, at the Answer. b al z2 + a2 2a And, a - the Answer, a 3. Reduce 5 to an improper fraction. 3a 4. Reduce 1 to an improper fraction.' Ans. 34. 3a Ans. 1 X 5. Reduce 2a 3ax + a2 to an improper fraction. 4x 18 6. Reduce 12 + to an improper fraction. 58 1 36 7. Reduce at to an improper fraction. 233 3a 8. Reduce 4 + 2x to an improper fraction, 5a CASE II. To Reduce an Improper Fraction to a Wholeor 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. EXAMPLES EXAMPLES. 16 ab + a? 1. To reduce and to mixed quantities: 3 b First, ist = 16 3 = 5j, the Answer required. ab + a2 And, -= ab + a2 b=a + Answer. 6 b 2ac 3a? 303 + 4.02 2. To reduce and to mixed quantities. a + 2ъс 3a2 322 First, 3a? C= 2a Answer. -= 3az + 4xOtx= 3x + - Ans, a + x a + x 33 2ar 3r2 13. Reduce and to mixed quantities. 5 322 Ans. 63, and 23 = lac с x2 42 2a? + 262 4. Reduce and to whole or mixed quan2a a - b tities. 3r2 2x3 2y3 5. Reduce and to whole or mixed x + y quantities. 10a2 40 + 6 6. Reduce to a mixed quantity. 50 15a3 + 5a2 7. Reduce to a mixed quantity. 3a3 + 2a2 2a-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 a az IZ с Here b bx Here and and --, by multiplying the terms of the IZ first fraction by z, and the terms of the 2d by x. 2 6 b 623 and ---, by multiplying the 6 ber bor бcx terms of the 1st fraction by bc, of the 2d by cx, and of the 3d by br. 2a 36 3. Reduce and to a common denominator. 20 4ac 36% Ans. and 2cx 2cx 2a 3a +26 4. Reduce and to a common denominator. 4ac 3ab + 262 2bc 5a 5. Reduce - and ---, and 4d, to a common denominator. 31 20 10ac 96x 24cda Ans. and and 6cc 6cx 60% 5 За 3a 6. Reduce and to fractions having a 6 6 206 18ab 486% + 720 common denominator. Ans. and and 246 246 2a2 + 62 7. Reduce - and and to a common denomi. 3 4 nator. 36 2c d 8. Reduce and and to a common denominator. 42? 3a 2a CASE 365, and 2b + 2a2 atb |