case of algebraic quantities are for the most part only indications of operations to be performed. All that we do is to transform the operations originally indicated into others, which are more simple, or which become necessary in order that the conditions of the question may be fulfilled. Thus, in the equation x + x + b = a given by the conditions of the question, art. 18, we simplify the operations originally indicated by reducing the expression x + x to one term, 2 x, by an operation analogous to addition in arithmetic, though not strictly the same. So likewise in question twentieth, art. 15, though we cannot, strictly speaking, subtract 576-12 x from 24 x, yet by an operation analogous to subtraction in arithmetic, we indicate upon these quantities operations, which produce the same effect, as the subtraction which the conditions of the question require. 31. Algebraic quantities consisting only of one term are called monomials, as 3 a, -4 b &c. Those which consist of two terms are called binomials, as a + b, c — d. Those which consist of three terms are called trinomials, &c. In general, quantities consisting of more than one term are called polynomials. Quantities consisting only of one term are also called simple quantities, and those consisting of more than one term are called compound quantities. Quantities in algebra, which are composed of the same letters, and in which the same letters are repeated the same number of times, are called similar quantities, thus, 3 a b, 7 a b are similar quantities, so also a a b, 5 a a b. 32. ADDITION OF ALGEBRAIC QUANTITIES. 1. Let it be required to add the monomials a, b, c, and d; the result, it is evident, will be a+b+c+d. 2. Let the quantities to be added be a b, c, a b, d. Here we have as before a b+c+ab+d; but the quantities a b, a b in this result are similar, they may therefore be united in one term, thus, 2 ab; whence the sum required will be 2 a b +c+d. To add monomials therefore, Write them one after the other with the sign+between them, observing to simplify the result by uniting in one, those which are similar. 3. Let it next be required to add the polynomials a + b and c+d+e. The sum total of any number of quantities whatever should be equal, it is evident, to the sum of all the parts of which these quantities are separately composed; we have therefore for the sum required a+b+c+d+e. Let the quantities proposed be a+b and c-d. If we begin by adding c, the result a + b + c will, it is evident, be too great by the quantity d, since it is not c, which we are to add, but c diminished by d; to obtain the true result therefore, from a+b+c we must subtract d; whence c- d added to a+b gives a+b+c-d. To add polynomials therefore, Write in order one after the other the quantities to be added with their proper signs, it being observed that the terms, which have no signs before them, are considered as having the sign +. 33. Let it next be required to add the following quantities. 9a+7b-2c By the rule just given the sum required will be 9a+7b-2c+2a-5c+8b+c In this result the similar terms 9a, 2a may be united in one sum 11 a; also the terms 7 b and 8b give 15 b. The similar quantities-2c,- 5 c being both subtractive, the effect will be the same, if we unite them in one sum 7 c and subtract this sum, and as there would still remain the quantity c to be added, instead of first subtracting 7 c and then adding c to the result, the effect will be the same if we subtract only 6 c. The sum of the expressions proposed will then be reduced to 11 a +15b -6 c. In order to verify this result, let us put numbers for the letters a, b, c, in the proposed; for example, the numbers, 10, 4, 3 respectively, we have 9a+7b-2c=112 2a-5c= 5 8b+c= 35 9a+7b-2c+2a-5c+8b+c=152 Making the same substitution in the expression 11a15b 6c, we obtain the same result. The operation, by which all similar terms are reduced to one, whatever signs they may have, is called reduction. To perform this operation, Take the sum of similar quantities, which have the sign + and that of those, which have the sign ; subtract the less of the two sums from the greater and give to the remainder the sign of the greater. We have then the following general rule for the addition of algebraic quantities, viz. Write the quantities in order one after the other with their proper signs, observing to simplify the result by reducing to one, terms which are similar. EXAMPLES. 1. To add the quantities 5a+3b-4 c 6c+2a-5b+2d 3e-4b-2c+a 7a-3c4b-6 e Answer 15a-2b-3c+2d-3e To verify this answer let the numbers 12, 5, 4, 3, 13, be put for the letters a, b, c, d, e, respectively. 2. To add the quantities 7m+3n-14 p+ 17 r 11n 26 m r+s Answer 31n-9m-9p+18r+3a-2b+s. 3. To add the quantities 11bc4ad-8ac+5cd 8 ac7bc-2 ad+4mn 9am- 2bc- 2 ad + 5 cd Answer 16 b c +5 ac + 12 c d +4mn-3ab+10 am. SUBTRACTION OF ALGEBRAIC QUANTITIES. 34. 1. To subtract a from b. Here the quantities being dissimilar, the subtraction can only be expressed by the sign thus, b a. 2. To subtract 5 a from 7 a. The quantities in this case being similar, the subtraction may be performed by means of the coefficients, and the result, it is evident, will be 2 a. 3. To subtract 2 b+3c from d. To subtract one quantity from another we must, it is evident, take from this other the sum of all the parts, of which the quantity to be subtracted is composed. The result required will therefore be d-2b-3 c. 4. To subtract a b from c. If we begin by subtracting a from c, it is evident, that we shall take away too much by the quantity b, by which a should be diminished before its subtraction; b should therefore be added to c - a to give the true result; whence ab subtracted from c gives c-a+ b. 5. To subtract 5 c +3 d. 4 b from 7c-2d-5b. The result, it is easy to see, will be 7 c — 2 d — 5 b -5c-3d+46 which becomes by reduction 2c-5 d b From what has been done the following rule for the subtraction of algebraic quantities will be readily inferred, viz. Change the signs + into, and the signs into in the quantities to be subtracted, or suppose them to be changed, and then proceed as in addition. EXAMPLES. + 1. To subtract from 17 a +2m-9b-4c+23 d the quantity Answer 51a-27b+11c-4d 2 m −34 a + 18 b — 15 c +27 d 2. To subtract from 5 a c-8 ab+9bc-4am the quantity Answer 8am-2ab+11ac-7 cd 9bc-6ac-6ab-12 am+7cd 3. To subtract from the quantity 15 abc-13 x y +21 cd-41 x — 25 75 xy-4abc+ 16 x 53 cd-31 m c Answer 19 abc-88 x y 57 x + 74 c d + 31 m c — 25 MULTIPLICATION OF ALGEBRAIC QUANTITIES. 35. 1. The product of a quantity a by another quantity b is expressed, as we have already seen, thus, a × b, or in a more simple manner, thus, a b. In like manner the product of ab by cd is expressed thus, a b c d, or thus, a b c d. 2. The letters a and b are called factors of the product a b. So also a, b, c and d are the factors of the product a b c d. The value of a product, it is easy to see, does not depend at all upon the order, in which its factors are arranged; thus the value of the product arising from the multiplication of a by b will evidently be the same, whether we write b a or a b. 3. Let it be proposed to multiply 3 a b by 5 cd; by no. 1 we have 3 ab 5 cd, or by no. 2, 3 × 5abcd; but the factors 3 and 5 in this result may, it is evident, be reduced to one by multiplying them together; performing this operation, the product required will be 15 a b c d. In like manner the product of the quantities 7 a b, 9 c d, 13 e f will be 819 a b c d e f. 4. Let it be required to multiply a a by a. According to no. 1 we have for the result a a a; but this expression for the product required may, it is easy to see, be abridged by writing the letter a but once only, and indicating by a figure the number of times this letter enters into it as a factor. The figure which indicates the number of times a given letter enters as a factor in a product is called the exponent of that letter. And in order to distinguish the exponent of a letter from a coefficient, we place the exponent at the right hand of the letter and a little above it, the coefficient being always placed before the letter, to which it belongs, and on the same line with it. According to this method the product a a is expressed by a2, a a a by a', a a aa by a*, &c. |