The first term a3 of the dividend divided by the first term a2 of the divisor, gives a2 for the first term of the quotient; multiplying the divisor, ab by the first term of the quotient, the result is a3a2b; subtracting a3-a2b from the dividend, the term as 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 ab-b3. Because the term a2b contains the letter a, we can divide it by the first term of the divisor, and we obtain ab, which is the second term of the quotient. Multiplying the divisor by +ab, the product is a b-ab2, which being subtracted from 2b-b2; the first term a2b destroys the term a2b which arose from the preceding operation; but there remains the term ab2, which being not yet in the dividend; the remainder is therefore ab2. 63. Dividing ab2 by a, the result is b2, 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+b2 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. * 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+ab+263 by a2+b2. Dividend. Divisor. 3 The first term-ab2 of the remainder, cannot be divided by a2, the first term of the divisor; thus the division terminates at this point. The fraction -ab2+b3 having the remainder for its numerator, a2+b2 and the divisor for its denominator, is annexed to the partial quotient a+b; and the complete quob3-ab2 tient is a+b+ 3 a2 +b2. 96. It is necessary to remark, that 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 first term of the divisor; as the succeeding part of the quotient, after this, would necessarily become fractional; and which may be carried on, ad infinitum, like a decimal fraction. This subject belongs to algebraic fractions, and as it is of considerable importance in analysis, we will treat of it with a near attention in the next Chapter. 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. |