CHAPTER XII. ON THE OPERATIONS OF MULTIPLICATION AND DIVISION IN mental as lical multi 566. THE fundamental assumptions which were made in FundaArt. 546, with respect to Symbolical addition and subtraction, sumptions are equally applicable, and for the same reasons, to Symbolical in symbomultiplication and division: they are as follows: 1st. Symbols which are general in form, are equally general and diviin representation and value. 2nd. The rules of the operations of multiplication and division in Arithmetical Algebra, when applied to symbols which are general in form though restricted in value, are applied without alteration to the operations bearing the same names in Symbolical Algebra, when the symbols are general in their value as well as in their form. It will follow from the second assumption that all the results of the operations of multiplication and division in Arithmetical Algebra, will be results likewise of Symbolical Algebra, but not conversely. plication sion. 567. The same three Cases of the operation of multiplication Three cases of symbolipresent themselves in Symbolical and in Arithmetical Algebra: cal multithey are as follows: 1st. When the multiplicand and multiplier are mononomials. 2nd. When the multiplicand is a polynomial and the mul tiplier a mononomial. 3rd. When both the multiplicand and multiplier are polynomials. plication. rence of like 568. In Arithmetical Algebra, the rule for the concurrence The rule for of like and unlike signs (Art. 57,) is required in the 2nd and the concur3rd Cases only: but in Symbolical Algebra, the occurrence symbols or single terms affected with the signs + and independently (Art. 544), renders its application necessary the three Cases under consideration. of and unlike signs reused quired in in all all the three cases. Its deduc tion from sumptions 569. In order to shew that the Rule of signs is a necessary the asconsequence of the assumptions made in Art. 566, we shall con- in Art. 566. VOL. II. C sider the product of a b and c d as determined by the principles of Arithmetical Algebra (Art. 56), which is Assuming, therefore, the permanence of this result, or in other words, the equivalence of the two members of which it is composed, for all values of the symbols, we may suppose two of their number to become successively equal to zero: thus, if we suppose b=0 and d=0, the product (1) in question becomes 1st. (a-0) (c− 0) = a c − a × 0 - 0 × c + 0 x 0, or a × b = ab, obliterating the terms which involve zero. If we suppose b=0 and c=0, we get, x-d--ad. 2nd. (a-0) (0–d) = a × 0-ad-0×0+0 × d, or a ×If we suppose a=0 and d= 0, we get, 3rd. (0-6) (c-0) = 0xc0×0-bc + b × 0, or -bxc=-bc. If we suppose a = 0 and c = 0, we get, 4th. (0-6) (0-d)=0x0-0xd - b × 0+ bd, or - bx-d=bd. It follows therefore generally, as a necessary consequence of the assumptions (Art. 566), which form the foundation of the The Rule. results of multiplication in Symbolical Algebra, that "when two like signs, whether + and + or and concur in multiplication, they are replaced in the product by the single sign +: and that when two unlike signs similarly concur, whether + and and +, they are replaced in the product by the single sign Rule for symbolical -> or 570. We now proceed to exemplify, in their order, the three multiplica- different Cases of Symbolical multiplication. tion for Case 1. CASE 1. When the multiplicand and multiplier are mono nomials. RULE. "In finding the mononomial product we must determine first, its sign; secondly, its coefficient; and lastly, its literal part." "Its sign is found by the rule for the concurrence of like and unlike signs which is deduced in the last article: when this sign is, it is commonly suppressed." "Its coefficient is the product of the coefficients of the several factors (Art. 37): if the coefficient be 1, (Art. 30), it is generally suppressed, as not necessary to be exhibited." "Its literal part is found by writing the several letters and their powers in immediate succession after each other, incorporating powers of the same letter into one by the rule given in Art. 41." It is proved in Arithmetical Algebra, (Art. 57), and therefore assumed in Symbolical Algebra, (Art. 566), that it is indifferent in what order the several component factors of a product succeed each other: it is on this account that it is usual to follow the alphabetical order, whenever the peculiar circumstances of the question under consideration do not render a departure from it convenient. 571. The following are examples: (1) ab=ab or + a ×+ b = +ab. (4) - axb- ab or - ax + b = − ab. These four examples merely express the rule for the concurrence of like and unlike signs. (Art. 569.) suppressing the number 4, which is common to the numerator and denominator. (Art. 76). 572. CASE 2. When the multiplicand is a polynomial and the multiplier a mononomial. Examples of Case 1. RULE. "Multiply the single term of the multiplier into Rule for every successive term of the polynomial, and arrange the pro- Case 2. ducts of the several terms in the result, preceded by their proper signs, in any order which may appear most symmetrical or most convenient." "If the sign of the mononomial multiplier be negative, the signs of all the terms of the multiplicand will be changed: in every other respect the rule agrees with that which is given in Arithmetical Algebra (Art. 50). Rule for Examples. 5x 3 574. CASE 3. When both the multiplicand and the multiplier are polynomials. RULE. "Multiply successively every term of one factor into every term of the other, add the several partial products together, and arrange the terms of the result in any order which may be considered most convenient, without regard to the sign of the first term." "If there be three factors, multiply the third into the product of the two first: and similarly for any number of them” (Art. 61). -- (4) (a3-b2+2b c − c3) ( − a3 + b2+2bc+ c2) = −a*+2a2b2 +2a2c2 - b*+2 b3 c2 - c1. (5) (x2+ ax + b) (x3 − a x + c) = x* — (a2 – b − c) x2 — (ab – ac) x +bc. 2 7ax3 107 a2x2 12 The examples of multiplication which are given in Arithmetical Algebra (Arts. 37, 51 and 69, and also Arts. 478 and 479) are examples likewise of Symbolical Multiplication, the processes followed being the same. symbol of reference. 576. It is usual to arrange the terms of the product according Letter or to the powers of some one letter, which may be called the letter or symbol of reference: thus x is the symbol of reference in Examples 1, 3, 5, 6, and a in Example 2: the same letter would be the symbol of reference in Example 4, if the result was reduced to the equivalent form − a* + 2 (b2 + c2) a3 — bˆ+2b3c2 — cˆ. The same result may be equally arranged under the equivalent forms - c*+2 (a2+b2) c2 - a* + 2 a2 b2- b', where b is the symbol of reference in one case, and c in the other. 577. The processes of division in Arithmetical and Sym- Division in Symbolical bolical Algebra merely differ in the additional rule which is Algebra. requisite for determining the sign of the quotient or of its first term, when a negative sign affects one or both of the first terms (or the only terms when both of them are mononomials) of the dividend and divisor. Cases of There are three Cases of the operation of Symbolical Division Three corresponding to the three cases of the operation of multipli- the opecation (Art. 567): we shall consider them in their order. ration of division. division when the divisor and 578. CASE 1. When both the dividend and divisor are Rule for mononomials. RULE. "The sign of the quotient is positive or negative, dividend according as the signs of the dividend and divisor are the same are monoor different." "The coefficient of the quotient is found by dividing the coefficient of the dividend by that of the divisor." "The literal part of the quotient is found by obliterating the symbols or their powers, which are common to the dividend and divisor and retaining those which are not thus suppressed." (Art. 78). nomials. |