5. Divide as-5a4b+10a3b2-10a2b3+5ab-b5 by a2 2ab+b2. Ans. a3-3a2b+3ab2-b 6. Divide 4823-96az3 -64a3z+150a3 by 2z-3a. 7. Divide b3b4x2+3b2x2 -x by b3-363x+3bx2-x3. 8. Divide a-x7 by α-x. 9. Divide a3+5a2x+5ax2+x3 by a+x. 10. Divide a+4a2b2 - 32b4 by a+26. 11. Divide 24a-b4 by 3a-26. 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 then the denominator being set under proper sign,+ or this sum, will give the improper fraction required. 1. Reduce 31, and 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. EXAMPLES. First, 163=51, the Answer required. ab+a+b=a+ Answer. b to mixed quantities. b ab+a3 And, b 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. first fraction by z, and the terms of the 2d by x. terms of the 1st fraction by bc, of the second by cx, and of the 3d by bx. CASE IV. To find the Greatest Common Measure of the Terms of a Fraction. DIVIDE the greater term by the less, and the last divisor by the last remainder, and so on till nothing remains; then the divisor last used will be the common measure required; just the same as in common numbers. But note, that it is proper to range the quantities according to the dimensions of some letters, as is shown in division. And note also, that all the letters or figures which are common to each term of the divisors, must be thrown out of them, or must divide them, before they are used in the operation. EXAMPLES. 1. To find the greatest common measure of ab+b2)ac2+bc2 or a+b Jac2+bc2 (c2® ac2+bca ab+b2 ac2+bca Therefore the greatest common measure is a+b. 2. To find the greatest common measure of a3-aba a2+2ab+ba Therefore a+b is the greatest common divisor. 3. To find the greatest common divisor of a2-4 ab+2b Ans. a-2, |