CHAPTER V. DIVISION. 69. Division by a monomial expression. have already considered the division of one monomial expression by another. We have also seen (Art. 43) that the quotient obtained by dividing the sum of two algebraical quantities by a third is the sum of the quotients obtained by dividing the quantities separately by the third; and we can shew by the method of Art. 54 that when any multinomial expression is divided by a monomial the quotient is the sum of the quotients obtained by dividing the separate terms of the multinomial expression by that monomial. Thus (a2x - 3ax) ÷ ax = a2x ÷ ax · 3αx÷ax = a — 3. And (12x3- 5αx2 – 2α2x) ÷ 3x - 2α2x÷3x=4x2 — §ax — §α2. = 12х3 + 3х - 5 ах2 ÷ 3х 70. Division by a multinomial expression. We have now to consider the most general case of division, namely the division of one multinomial expression by another. Since division is the inverse of multiplication, what we have to do is to find the algebraical expression which, when multiplied by the divisor, will produce the dividend. Both dividend and divisor are first arranged according to descending powers of some common letter, a suppose; and the quotient also is considered to be so arranged. Then (Art. 62) the first term of the dividend will be the product of the first term of the divisor and the first term of the quotient; and therefore the first term of the quotient will be found by dividing the first term of the dividend by the first term of the divisor. If we now multiply the whole divisor by the first term of the quotient so obtained, and subtract the product from the dividend, the remainder must be the product of the divisor by the sum of all the other terms of the quotient; and, this remainder being also arranged according to descending powers of a, the second term of the quotient will be found as before by dividing the first term of the remainder by the first term of the divisor. If we now multiply the whole divisor by the second term of the quotient and subtract the product from the remainder, it is clear that the third and other terms of the quotient can be found in succession in a similar manner. For example, to divide 8a3 + 8a3b + 4ab2 + b3 by 2a + b. 2a + b) 8a3 + 8a3b + 4ab2 + b3 ( 4a2 + 2ab + b2 4a2b+ 4ab2 + b3 4a2b+2ab2 2ab2 + b3 = 4a2. The first term of the quotient is 8a3 ÷ 2a Multiply the divisor by 4a2 and subtract the product from the dividend: we then have the remainder 4ab + 4ab2 + b3. The second term of the quotient is 4ab2a = 2ab. Multiply the divisor by 2ab, and subtract the product from the remainder: we thus get the second remainder 2ab2+b3. The third term of the quotient is 2ab2 ÷ 2a = b2. Multiply the divisor by b2, and subtract the product from 2ab2 + b3, and there is no remainder. Since there is no remainder after the last subtraction, the dividend must be equal to the sum of the different quantities which have been subtracted from it; but we have subtracted in succession the divisor multiplied by 4a, by + 2ab, and by + b2; we have therefore subtracted altogether the divisor multiplied by (4a2+2ab+b2). And, since the divisor multiplied by 4a2+2ab+b is equal to the dividend, the required quotient is 4a2 + 2ab + b2. The dividend and divisor may be arranged according to ascending instead of according to descending powers of the common letter, as in the last example considered with reference to the letter b; but the dividend and the divisor must both be arranged in the same way. 71. The following are additional examples: Ex. 1. Divide aa — a3b +2a2b2 – ab3+b1 by a2+b2. a2+b2) a1 — a3b+2a2b2 — ab3+ba ( a2 − ab + b2 a4 +a2b2 The In this example the terms of the dividend were placed apart, in order that 'like' terms might be placed under one another without altering the order of the terms in descending powers of a. subtractions can be easily performed without placing 'like' terms under one another; but the arrangement of the terms according to descending (or ascending) powers of the chosen letter should never be departed from. Ex. 3. Divide a3 + b3 + c3 – 3abc by a+b+c. a+b+c) a3 − 3abc + b3 + c3 ( a2 − ab − ac + b2 − bc+c2 a3 + a2b+a2c Where, as in the above example, more than two letters are involved, it is not sufficient to arrange the terms according to descending powers of a; but b also is given the precedence over c. By using brackets, the above process may be shortened. Thus a+b+c) a3 - 3abc + b3 + c3 ( a2 − a (b+c) + (b2 − bc + c2) a3 + a2 (b+c) - a2 (b+c) - 3abc + b3 + c3 - a2 (b + c ) − a (b+c)2 a (b2 - bc+c2)+b3+c3 a (b2 - bc+c2)+b3+c3 72. The method of detached coefficients may often be employed in Division with great advantage. For example, to divide 2x-7x5+5x+3x3-3x2 + 4x-4 by 2x3-3x2+x-2, we write 2-3+1-2)2-7+5+3-3+4-4(1-2-1+2 2-3+1-2 -445-3+4-4 -46-2+4 -2+7-7+4-4 -2+3-1+2 4-6+2-4 4-6+2-4 The first term of the quotient is a and the other powers follow in order: thus the quotient is 73. Extended definition of Division. In the process of division as described in Art. 70, it is clear that the remainder after the first subtraction must be of lower degree in a than the dividend; and also that every remainder must be of lower degree than the preceding remainder. Hence by proceeding far enough we must come to a stage where there is no remainder, or else where there is a remainder such that the highest power of a in it is less than the highest power of a in the divisor, and in this latter case the division cannot be exactly performed. It is convenient to extend the definition of division to the following: To divide A by B is to find an algebraical expression C such that B x C is either equal to A, or differs from A by an expression which is of lower degree, in some particular letter, than the divisor B. For example, if we divide a2 + 3ab + 4b2 by a+b, we have a+b) a2+3ab + 4b2 ( a + 2b 2ab+2b2 +262 Thus (a+3ab + 4b2) ÷ (a + b) = a + 2b, with remainder 262; that is a2+3ab + 4b2 = (a+b) (a + 2b) + 262. We have also, by arranging the dividend and divisor differently, Hence a change in the order of the dividend and divisor leads to a result of a different form. This is, however, what might be expected considering that in the first |