RULE. By Case I. find the sum of the coefficients of the positive quantities, and the sum of the coefficients of the negative quantities; then to the difference of these sums prefix the sign of the greater sum, and annex the common literal quantity. 7a, 11a, 6a, we get Thus, to find the sum of 9a, - 7a, 11a, = = for the sum of the positive quantities 20a, and that of the negative quantities - 13a, and 13a taken from 20a gives 7a. It is evident that our result is correct, for a is to be taken as often as there are units in the difference of the sums of the coefficients of the positive and negative quantities, and it must clearly have the sign of the greater sum. (4.) When the quantities are not all similar. RULE. By Cases I. and II. add all the like quantities of any one sort, and to this result set down the quantities which remain, one after the other, with its proper sign. It is evident (by Ax. I.) that it is of no consequence in what order the quantities are placed with regard to each other, either in uniting the similar quantities of any one sort into one, or in annexing the remaining quantities. To add 5a3x + 9b3y - 132 v3 + mn, 14a3x + 2063y + 3z3v3 + r — 3, 11b3y — 6b3y + w — 3t + 7z2v3, c2 −ď2 + √p− Vq, we place the like quantities of any one sort under each other, and we get 19ax +34b3y - 3z2v3 + c2—d2 + m—n+r−3+w—3t+ √p 13 √x2 + y2+ 5 √x2 — y2 + 14pq + 7d + 9c + 11w 2 √ x2 + y2 + 2 √ x2 − y2 + 9pq + √μ3 + 20†Ãa + (lm)* (5.) If the coefficients are literal, then their sum may be found as above; and the sum thus found, being written within a parenthesis, which precedes the common literal quantity, will represent the sought sum. Thus, ax+by+ cz1 + mxy, rx2 — ly3 + pz1 — nxy, 3x2 py3 + qz1 — 5mxy + w, 5ex2 + 11m3y3 — 14ƒ1⁄2* + 2mxy — PQ, when properly written and added, give sum = (a+r+3+5e)x2+(b− 1 − p +11m2)y3 + (c+p+q− 14f)2- (2m + n)xy + w — pq as required; for evidently a2 is to be taken as often as there are units in the sum of a, r, 3, 5e; and y3 is to be used as often as there are units in the sum of b, — l, — p, + 11m3, and so of the rest. - (1.) SUBTRACTION consists in taking one quantity, called the subtrahend, from another quantity of the same kind, called the minuend, and in finding a suitable expression for the result, which is termed the difference or remainder. b Thus, if a stands for the sum of all the positive terms of the minuend, and 6 for the sum of all the negative terms, then ab will stand for the minuend; also, if c denotes the sum of all the positive terms of the subtrahend, and — d denotes the sum of all its negative terms, then cd will represent the subtrahend, and a -b - (ed) will indicate the subtraction of c d from a - b. We observe since c is greater than cd by d, that if we subtract e from ab, the remainder will be too small by d; therefore, by adding d, we shall have the true remainder. Now, c subtracted from a ―b, gives a b c for the remainder, and adding d to this remainder, we get ab− c + d for the true remainder, after subtracting cd from a - b. It is easy to see that in subtracting cd from ab, we have changed the signs of c-d, and added the result to a-b; for cd, by changing its signs, becomes c+d, which, added to ab, gives a-b-c+d, as above. Indeed, since Subtraction is the reverse of Addition, we may evidently regard the minuend as being equal to the sum of the subtrahend and remainder. Hence, if we change the signs of all the terms of the subtrahend, and add it to the minuend, we shall have for the result, the remainder + the subtrahend the subtrahend the remainder; since by Axiom VI. the subtrahend - the subtrahend = 0 (or naught). (2.) From what has been said, we deduce the following rule for Subtraction: RULE. Change the sign of each term of the subtrahend; that is, change each into, and each into +; or, which is better in practice, conceive the signs to be changed; and then add the subtrahend, with its signs supposed to be changed, to the minuend, by the rules given in Addition. The resulting sum will be the remainder or difference required. -2√a+b+ 9 √x2-y3-50pq - 78mn 3√a1 + b1 − 16 √x3 − y3 + 49b3c1 + 50pq — 11efg + 78mn 5. n (3.) From what has been done, it is manifest that we may write the polynomial abcdefghk - linn, in the form abcdef - (lmn-ghk), or in the form (def + Imn — abc ghk). In like manner other polynomials may be written in different forms, observing that the sign before the parenthesis is used to signify that the quantity within the parenthesis is to be subtracted, or that its signs are to be changed. It is to be noted that we have used the word polynomial to denote any quantity that consists of more than one term. REMARKS ON ADDITION AND SUBTRACTION. (1.) It is easy to see, from what has been done in Addition, that the sum of any number of quantities in Algebra is not of necessity greater than either of the quantities added, and equal to their absolute sum, as in Arithmetic. The reason is plain; for in Arithmetic the quantities added are all supposed to be positive, whereas in Algebra some of the quantities added may be positive and others negative, so that, in forming the sum, the quantities are to be united |