Ex. 2. Divide a-b3 by a-b. • Division MULTIPLICATIOy. Dividend. Divisor. Mul. a -a3 63 a-6 by a? Hab +6° 93-a2b Quotient. al-a'b ab-bas tab+62 ta2b-aba *a*baba taba--6% 等 a3 -b? ab? -63 * The first term a?. of the dividend divided by the first term a’ of the divisor, gives a for the first term of the quotient; 'multiplying the divisor az by al? the first term of the quotient, the result is a ab; subtracting a3-a6 from the dividend, the term a® destroys the first term of the dividend, but there remains the term -a2b, which is not found at first in the dividend; therefore the remainder is b-b3. Because the term ab contains the letter a, we can divide it by the first term of the divisor, and we obtain tab, which is the second term of the quotient. Multiplying the divisor by +ab, the product is a2b--aba, which being subtracted from a2b_b2 ; the first term aob destroys the term ao b which arose from the preceding operation; but there remains the term ab', which being not yet in the dividend; the remainder is therefore ab? 63. Dividing ab? by a, the result is ba, which is the third term of the quotient; multiplying the divisor by be, we have ab-b3; and subtracting this result from the last remainder, the terms of both destroy one another; so that nothing remains In order to comprehend well the mechanism of the division, it is only necessary to take a glance at the multiplication of the quotient a2 + ab +62 by the divisor a--b, and it will be readily seen that all the terms reproduced in the partial divisions are those which destroy one another in the result of the multiplication. Ex. 3. Divide yi-1 by y--1. Dividend. Divisor. a-X Ex. 4. Divide a6-206 by a-X. Dividend. Divisor. Quotient. a4 x 2 -a2x4 4 3 -a 6 a3x3 Ex. 5. Divide 5+-as by sta. Dividend. Divisor. 75 tas a ta x5 fax4 Quotient. axtas xaxta.co -axta -ax4-a2x3 95. When we apply the rule, (Art. 93), to the division of algebraic quantities of which one is not a factor of the other, we know it is impossible to effect the division; because that we arrive, in the course of the operation, at a remainder of which the first term cannot be divided by that of the divisor. In this case, the remainder is made the numerator of a fraction whose denominator is the divisor; and the fraction thus arising, with its proper sign, is annexed to the other part of the quotient, in order to render its value complete. Ex. 6. Divide a3 +a2b+263 by a2 +62. Dividend. Divisor. Quotiento 63 -- aba aab +63 a+b+ 2 +62 3 The first term-aba of the remainder, cannot be divided by a”, the first term of the divisor ; thus the division terminates at this point. The fraction -ab? +63 having the remainder for its numerator, al +62 and the divisor for its denominator, is annexed to the partial quotient a+b, and the complete quo 63ab2 tient is a+b+ a® +62 97. In the preceding examples, the product of the first term of the quotient by the divisor, is placed under the dividend; then the reduction is made by subtraction; and every succeeding product is managed in like manner. In the following examples, the signs of all the terms of the product are changed in placing it under the dividend ; and then the reduction is performed by the rules of addition; which is the method adopted by some of the most refined Analysts. Ex. 7. Divide a + 2a'ba +-64-c" by a+12+02: Dividend. Divisor, ast-2a2b2 +64 ---24 a2+bo+c? -24 a2b2 --ac? Quotient. Ist rem. a ba?c? +64--4 a+62 - --a262-62c2-64 ?d rem. -aac2 -62 c2 taca +62 ca toc * Ex. S. Divide 6x4-96 by 3x -6. Divisor. 3—6 Quotient. +242-96 48x---96 482-96 Es. 9. Divide 8a® --- 4aRb2 + 423 --- 2a3 ---+1 . by 2a-b2 +1. Dividend. Divisor. Sab-4a3b2 +-4a" +-2a-b2 +12a-b2+1 --8a® +1a3b2-4a3 Quotient. 4a3+1 2a3--62 +1 -2a3 +62 - 1 * |