The coefficients of the remaining terms of the multiplier being placed either in curve to the right, as “(— 3 + 2," or in any other way that may be thought convenient. It may be further remarked, for the purpose of connecting the identity of arithmetic, in its usual form, with the practice of algebra, in the student's mind, that if all the signs were + (for this is always so considered in arithmetic) and we make a 10, the above process would be precisely the same as is practised in ordinary multiplication, except that the order is inverted. Note I. In the multiplication of compound quantities, it is usually best to set them down in order, according to the powers and the letters of the alphabet. And in the actual operation, begin at the left-hand side, and multiply from the left-hand towards the right, in the manner that we write, which is contrary to the usual way, though analogous to the Indian, of multiplying numbers. But in setting down the several products, as they arise, in the second and following lines, range them under the like terms in the lines above, when there are such like quantities; which is the easiest way for adding them together. In many cases, the multiplication of compound quantities need only be indicated by setting them down one after another, each within or under a vinculum; and either with a sign of multiplication between them, as (a + b) × (a − b) × 3ab, or in juxtaposition, (a + b) (a - b) 3ab. Note II. The operations in multiplication will often be greatly facilitated, by fixing the following rules and formula well in the recollection. The square of any polynomial is equal to the sum of the squares of its terms + twice the product of every two of its terms taken in all their different combinations. Thus, (a + b + c + d) (a + b + c + d) = a2 + b2 + c2 + d2 +2ab+2ac + 2ad and (a+b+c+ d + e + f) (a + b + c + d + e +f) =a2 + b2 + c2 + ď2 + e2 + ƒ2 +2ab+2ac + 2ad + 2ae +2af +2bc + 2bd + 2be + 2bf +2cd2ce + 2cf + 2de + 2df In all such cases the arrangement of the products is very simple, and the continuation of the process very obvious. Note III. From the principle that the rectangle of the sum and difference of two quantities is equal to the difference of their squares, some useful theorems obviously flow: viz. 1. a2 — b2 = (a + b) (a − b ). 2. a4 -- · b1 = (a2 + b2) (a2 — b2) = (a2 + b2) (a + b) (a (a + b1) (a1 — b1) = (a1 + b1) (a2 + b2) (a + b) (a — b). (a3 + b2) (a + b1) (a2 + b2) (a + b) (a - b). (a2 + ab + b2) (a — b). 9. (x + a) (x + b) (x + c) = x3 + (a + b + c) x2 + (ab + ac + bc) x + abc. 6. - 7x2. 3x2y2 + 3y3. 2b) x. X. 9. Multiply 323 + 2x2y2 + 3y3 by 2x3 10. Multiply (a2 + ab +b2) y by (a 11. Multiply a” + aTM−1 x + aTM−2 x2 + 12. Multiply ax + bx2 + cx3 by 1 + x + x2 + x3, and consider a, b, c, as coefficients of the powers of a: as in p. 120. .... a) (x 13. Develope (x + a) (x + b) (x + c) (x + d) and also (x b) (x —c) (x-d); and attach the combinations of a, b, c, d, to the powers of x as coefficients. 14. The student may take in another factor (≈ + e), and it would be worth his while to attempt to discover inductively some law or rule by which he could form the terms seriatim, without the trouble of writing down the previous steps. 15. Multiply x2 + (a - b) x — ab by x2 + (c 16. (1+r+ p2 + p3 ́. , g") × (1 d) x r) = DIVISION. DIVISION in algebra, like that in numbers, is the converse of multiplication; and it is performed like that of numbers also, by beginning at the left-hand side, and dividing all the parts of the dividend by the divisor, when they can be so divided; or else by setting them down like a fraction, the dividend over the divisor, and then abbreviating the fraction as much as can be done, by cancelling any quantities which are common both to numerator and denominator. This may naturally be distinguished into the following particular cases. CASE I. When the divisor and dividend are both simple quantities. SET the terms both down as in division of numbers, either the divisor before the dividend, or below it, like the denominator of a fraction. Then abbreviate these terms as much as can be done, by cancelling or striking out all the letters that are common to them both, and also dividing the one coefficient by the other, or abbreviating them after the manner of a vulgar fraction in arithmetic, by dividing them by their common measure. It will, of course, be necessary to subtract the index of the less power (whether it be in the numerator or denominator of the fraction thus formed) from the index of the greater, leaving the difference where the greater index previously was. If, however, on the contrary, any ulterior purposes render it advantageous (and this often happens in algebraic investigations) to keep the latter in that term of the fraction, from which it would thus be expunged, we may subtract the greater index from the less, and put the difference with the sign Thus b2 b2a-3 (that is b2a-3), or a3, a2 bз or 1 1 or -. And a-3 may be written either a5 b a3 b-2 b-2 so with any other quantities which appear in the result of the indicated division. Note. Like signs in the two factors make + in the quotient; and unlike -; the same as in multiplication *. signs make When the dividend is a compound quantity, and the divisor a simple one. DIVIDE every term of the dividend by the divisor, as in the former case. * Because the divisor multiplied by the quotient must produce the dividend. Therefore, 1. When both the terms are +, the quotient must be +; because in the divisor multiplied by in the quotient, produces + in the dividend. 2. When the terms are both by the quotient is also +; because in the quotient, produces + in the dividend. 3. When one term is + and the other, the quotient must be multiplied by - in the quotient produces in the dividend, or by in the quotient gives — in the dividend. in the divisor multiplied ; because in the divisor in the divisor multiplied So that the rule is general; viz. that like signs give +, and unlike signs give-, in the quotient. 12. Find the quotient of 2x3-5 + y2n−3 by 6x1—5m y−1. 14 16 13. Divide a+a — a by 5a3b2. 14. Divide ·001ä2 + 1000μ3 + ·01-01 by '6æ ̄3; and 1031⁄23 + 3.103y2 + 5 × 102xy by '06 × 103Ã3. (a + b + c)3. 16. Divide 00015 × 105 +03 × 103r by 0005-46; and 00015-2 × 10−5a3 - 6-4 × 10-3 x 001-2 by 10-1. When the divisor and dividend are both compound quantities. 1. Set them down as in division of numbers, the divisor before the dividend, with a small curved line between them, and range the terms according to the powers of some one of the letters in both, the higher powers before the lower. 2. Divide the first term of the dividend by that of the divisor, as in the first case, and set the result in the quotient. 3. Multiply the whole divisor by the term thus found, and subtract the result from the dividend. 4. To this remainder bring down as many terms of the dividend as are requisite for the next operation, dividing as before; and so on to the end, as in common arithmetic. Note I.. If the divisor be not exactly contained in the dividend, the quantity which remains after the operation is finished may be placed over the divisor, like a vulgar fraction, and set it down at the end of the quotient, as in arithmetic. Note II. By observing that the number of terms in any remainder that takes place after all the terms are brought down from the dividend is always less than the number of terms in the divisor, it is clear that, however far the operation is carried, the work can never terminate. The remainder always occurring, the terms of the quotient may always be increased; and that without any assignable limit. The series of terms thus formed is, from its capability of unlimited extension, called an INFINITE SERIES. By attending to the manner in which the successive terms are related to the preceding one or preceding ones, the law of the progression (in Infinite Series resulting from Division) may be always and very readily discovered: so that when a few of the first terms have been actually obtained by the prescribed process, the remaining ones may be written out to any extent we may choose or require, by merely attending to this law of observed dependence. Examples of these will be found under the head of Infinite Series in a future part of this volume. |