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, 3a2-262-4ab, 5a2-b2+2ab, and 3ab-3c2-2b2. The term 3a2 being similar to 5a2, we write 8a2 for the result of the reduction of these two terms, at the same time slightly crossing them, as in the first term. 342-4ab-263 -5a2+2ab- b3 +3ab-26-3c2 8a+ ab-5b2 — 3c2 Passing then to the term -4ab, which is similar to +2ab and +3ab, the three reduce to +ab, which is placed after 8a3, and the terms crossed like the first term. Passing then to the terms involving b3, we find their sum to be—562, after which we write —3c2. The marks are drawn across the terms, that none of them may be overlooked and omitted. 4. Add together the polynomials 5a2b+6cx+9bc2, 7cx—8a2b+ Va and -15cx-9bc2+2a2b. Ans. √a-ab-2cx. 5. Add together the polynomials √x+ax—ab, ab— √x+xy, ax+xy-4ab, √x + √x-x and xy+xy+ax. Ans. 2√x+3ax-4ab+4xy-x. 6. Add together the polynomials 15axy+5bc2+3aƒ2, 3af2+ √xÿ -12xay,-5bc2+√ay-3axy, and -2√ay - √x-baƒ3. 6af2. Ans. √xy-ay-√x. 7. Add together the polynomials 7a2b-3abc-8bc-9c2+cd3, Sabc-5ab+3c3-463c+cd2 and 4a2b-8c3+9bc-3d3. 36. Subtraction, in algebra, consists in finding the simplest expression for the difference between two algebraic quantities. The result obtained by subtracting 46 from 5a is expressed by 5a-4b. In like manner, the difference between 7a3b and 4a3b is expressed by 7a3b-4a3b=3a3b. Let it be required to subtract from 4a 2b-3c In the first place, the result may be written thus, 4a-(2b-3c) by placing the quantity to be subtracted within the parenthesis, and writing it after the other quantity with the sign. But the question frequently requires the difference to be expressed by a single polynomial; and it is in this that algebraic subtraction principally consists. To accomplish this object, we will observe, that if a, b, c, were given numerically, the subtraction indicated by 26-3c, could be performed, and we might then substract this result from 4a; but as this subtraction cannot be effected in the actual condition of the quantities, 26 is subtracted from 4a, which gives 4a-2b; but in subtracting the number of units contained in 2b, the number taken away is too great by the number of units contained in 3c, and the result is therefore too small by the same quantity; this result must therefore be corrected by adding 3c to it. Hence, there results from the proposed subtraction 4a-2b+3c. The difference is expressed by 8a2-2ab-(5a2-4ab+3bc—b3) The reduction is made by observing, that to subtract 5a2-4ab +3bc-b2, is to subtract the difference between the sum of the additive terms 5a2+3bc, and the sum of the substractive terms 4ab+b2. We can then first subtract 5a2+3bc, which gives 8a2-2ab-5a3 -3bc; and as this result is necessarily too small by 4ab+, this last quantity must be added to it, and it becomes 8a2-2ab-5a2 -3bc+4ab+b2; and finally, after reducing, 3a2+2ab-3bc+b2. 37. Hence, for the subtraction of algebraic quantities, we have the following general RULE. I. Write the quantity to be subtracted under that from which it is to be taken, placing the similar terms, if there are any, under each other. II. Change the signs of all the terms of the polynomial to be subtracted, or conceive them to be changed, and then reduce the polynomial result to its simplest form, 7. From 8abc-12b3a+5cx-7xy, take 7cx-xy-13b'a. Ans. 8abc-b3a-2cx-6xy. 38. By the rule for subtraction, polynomials may be subjected to certain transformations. —: they are very These transformations consist in decomposing a polynomial into two parts, separated from each other by the sign useful in algebra. 39. REMARK.-From what has been shown in addition and subtraction, we deduce the following principles. 1st. In algebra, the words add and sum do not always, as in arithmetic, convey the idea of augmentation; for a-b, which results from the addition of -b to a, is properly speaking, a difference between the number of units expressed by a, and the number of units expressed by b. Consequently, this result is less than a. To distinguish this sum from an arithmetical sum, it is called the algebraic sum. Thus, the polynomial 2a-3a2b+36'c is an algebraic sum, so long as it is considered as the result of the union of the monomials 2a3, -3ab, +36'c, with their respective signs; and, in its proper acceptation, it is the arithmetical difference between the sum of the units contained in the additive terms, and the sum of the units contained in the subtractive terms. It follows from this that an algebraic sum may, in the numerical applications, be reduced to a negative number, or a number affected with the sign 2d. The words subtraction and difference do not always convey the idea of diminution, for the difference between a and b being a+b, exceeds a. This result is an algebraic difference, and can be put under the form of a-(-b). MULTIPLICATION. 40. Algebraic multiplication has the same object as arithmetical, viz. to repeat the multiplicand as many times as there are units in the multiplier. It is generally proved, in arithmetical treaties, that the product of two or more numbers is the same, in whatever order the multiplication is performed; we will, therefore, consider this principle demonstrated. This being admitted, we will first consider the case in which it is required to multiply one monomial by another. The expression for the product of may at once be written thus 7a3b2 by 4a2b But this may be simplified by observing that, from the preceding principles and the signification of algebraic symbols, it can be written 7x4aaaaabbb. Now, as the co-efficients are particular numbers, nothing prevents our forming a single number from them by multiplying them together, which gives 28 for the co-efficient of the product. As to |