RULE. 1. Form a single additive term of all the terms preceded by the sign plus: this is done by adding together the co-efficients of those terms, and annexing to their sum the literal part. II. Form, in the same manner, a single subtractive term. III. Subtract the less sum from the greater, and prefix to the result the sign of the greater. REMARK. It should be observed that the reduction affects only the co-efficients, and not the exponents. EXAMPLES. 1. Reduce the polynomial 4a2b-8a2b-9a2b+11a2b to its simplest form. Ans. -2a2b. 2. Reduce the polynomial 7abc-abc-7abc2-8abc2+6abc2 to its simplest form. Ans. -3abc2. 3. Reduce the polynomial 9cb-8ac2+15cb3+8ca+9ac2-24cb to its simplest form. Ans. ac2+8ca. The reduction of similar terms is an operation peculiar to algebra. Such reductions are constantly made in Algebraic Addition, Subtraction, Multiplication, and Division. 30. It has been remarked in Definition 3, that the quantities considered in algebra are represented by letters, and the operations to be performed upon them, are indicated by signs. The letters and signs are used to abridge and generalize the reasoning required in the resolution of questions. 31. There are two kinds of questions, viz. theorems and problems. If it is required to demonstrate the existence of certain properties relating to quantities, the question is called a theorem; but if it is proposed to determine certain quantities from the knowledge of others, which have with the first known relations, the question is called a problem. The given or known quantities are generally represented by the first letters of the alphabet, a, b, c, d, &c. and the unknown or required quantities by the last letters, x, y, z, &c. 32. The following question will tend to show the utility of the alge. braic analysis, and to explain the manner in which it abridges and generalizes the reasoning required in the resolution of questions. Question. The sum of two numbers is 67, and their difference 19; what are the two numbers? Solution. We will begin by establishing, with the aid of the conventional signs, a connexion between the given and unknown numbers of the question. If the least of the two required numbers was known, we would have the greater by adding 19 to it. This being the case, denote the least number by : the greater may then be designated by x+19: hence their sum is x+x+19, or 2x+19. But from the enunciation, this sum is to be equal to 67. There. fore we have the equality or equation 2x+19=67. Now, if 2x augmented by 19, gives 67, 2x alone is equal to 67 minus 19, or 2x=67-19, or performing the subtraction, 2x=48. Hence x is equal to the half of 48, that is, x==24. The least number being 24, the greater is x+19-24+19=43. And indeed, we have 43+24=67, and 43-24=19. Table of the Algebraic Operations. Hence, 2x+19=67, and 2x=67-19; therefore x==24 and consequently x+19=24+19=43, And indeed, 43+24=67, 43-24-19. Another Solution. Let a represent the greater number, a-19 will represent the least. Hence, 2x-19=67, whence 2x=67+19; therefore, and consequently, x==43 x-19-43-19-24. From this we see how we might, with the aid of algebraic signs, write down in a very small space, the whole course of reasoning which it would be necessary to follow in the resolution of a problem, and which, if written in common language, would often require several pages. General Solution of this Problem. The sum of two numbers is a, their difference is b. What are the two numbers? As the form of these two results is independent of any particular value attributed to the letters a and b, it follows that, knowing the sum and difference of two numbers, we obtain the greater by adding the half difference to the half sum, and the less, by subtracting the half difference from half the sum. Thus, when the given sum is 237, and the difference 99, And indeed, 168+69=237, and 168-69=99. From the preceding question we perceive the utility of repre senting the given quantities of a problem by letters. As the arithmetical operations can only be indicated upon these letters, the result obtained, points out the operations which are to be performed upon the known quantities, in order to obtain the values of those required by the question. a b The expressions 이 obtained in this prob lem, are called formulas, because they may be regarded as comprehending the solutions of all questions of the same nature, the enunciations of which differ only in the numerical values of the given quantities. Hence, a formula is the algebraic enunciation of a general rule. From the preceding explanations, we see that Algebra may be regarded as a kind of language, composed of a series of signs, by the aid of which we can follow with more facility the train of ideas in the course of reasoning, which we are obliged to pursue, either to demonstrate the existence of a property, or to obtain the solution of a problem. ADDITION. 33. Addition, in Algebra, consists in finding the simplest equivalent expression for several algebraic quantities, connected together by the sign plus or minus. Such equivalent expression is called their sum. 34. Let it be required to add together the expressions. The result of the addition is За 56 2c an expression which cannot be reduced to a more simple form. Let it be required to find the sum of the expressions. { Their sum, after reducing (Art. 29), is .. 3a2-4ab 2ab-56 5a2-5ab-46 35. As a course of reasoning similar to the above would apply to all polynomials, we deduce for the addition of algebraic quantities the following general RULE. I. Write down the quantities to be added so that the similar terms shall fall under each other, and give to each term its proper sign. II. Reduce the similar terms, and annex to the results, those terms which cannot be reduced, giving to each term its respective sign. EXAMPLES. 1. Add together the polynomials, 3a2-2-4ab, 5a2-b2+2ab, and 3ab-3c2-2. The term 3a2 being similar to 5a2, we write 8a for the result of the reduction of these two terms, at the same time slightly crossing them, as in the first term. 342-4ab-262 +3ab-2-3c2 8a2+ ab-5-3c2 Passing then to the term -4ab, which is similar to +2ab and +3ab, the three reduce to +ab, which is placed after 8a", and the terms crossed like the first term. Passing then to the terms involving b, we find their sum to be-56, after which we write -3c2. The marks are drawn across the terms, that none of them may be overlooked and omitted. (2). 7x+3ab+2c -3x-3ab- 5c 5x-9ab-9c Sum. 9x-9ab-12c (3). 9a2b+bc+ abc 3a2b2-7bc+5abc. |