which is negative; indeed, since ab is b times as great as a, ab is b times as great as it is manifest that -ax b: = ab. -b a, or that 3. If a and b represent the multiplicand and multiplier, a being positive and negative, then if e is a positive. number or quantity of the same kind as b, we evidently have − b = c − (b + c); and of course the multiplication of a by -b is reduced to the multiplication of a by c− (b + c). Because c equals the sum of e- (b + c) and b + c, it follows that we must multiply a by c, and then from the product subtract the product of a and b + c; consequently, we shall have a x bax [c (b+c)] = ac a(b + c). c And since ab+c) = a + a + a +, etc., to (b+c) terms, = a + a +, etc., to b terms, + a +a+, etc., to c terms, = abac, it follows that ac a(b + c) = ac — ab - ac or (since acac = 0) = ab; consequently, a x − b = — ab, a negative product. 11 4. Let a and b, regarded as negative, represent the multiplicand and multiplier; or, as has just been shown, they may be represented by a and c - (b+c). Because c exceeds the units in c (b+c) by the units in b+c, we must multiply -a by c, and subtract the product of a and b + c from the result; consequently (agreeably to what has been done), we shall have - ax-b= [− a(b + c)] ac + ab + ac, or (since we shall have a bab, which is a positive product. Hence, when the multiplicand and multiplier have like signs, their product is positive; and when the multiplicand and multiplier have unlike signs, their product is negative. It is hence evident that the product of any number of factors is positive when it does not contain an odd number of negative factors, and when an odd number of negative factors enter the product it is clearly negative. (10.) Although the algebraic quantities that may be used in multiplication may indefinitely represent either integral, fractional, or irrational quantities, yet we shall multiply them as if they were integral, because (as we shall hereafter prove) it is allowable to do so. (11.) From what has been done, we may evidently derive the following rule for the multiplication of any two monomials together. RULE. To the product of the coefficients of the monomials annex all the letters or literal quantities that belong to each, with their proper exponents. Then if the same letter or literal quantity occurs more than once, write the letter or literal quantity once, with an exponent equal to the sum of the ехроnents of the letter or literal quantity, added according to their signs; observing that if no exponent is expressed, it is understood to be 1. If the monomials are both positive or both negative, the product will be positive; but if one of them is positive and the other negative, the product is negative; in other terms, like signs give +, and unlike signs give, for the sign of the product. REMARK. If the monomials have no coefficients expressed, the product of the coefficients is 1; for when any quantity has no coefficient expressed, it is always understood to be 1. Thus, the product of 7ab and 5ac is 7ab x 5ac = 7 x 5aabc35a1a1bc35abc, which is positive, because the monomials are positive. - Also, the product of 3abe and 7abcd is - 3a2bc × -7abcd21a ab1bc'c'd = 21abcd, the product being positive, because the signs of the monomials are alike. Again, the product of9abc and 15abdefg is 135abcdefg, the product having the sign —, because the monomials have unlike signs. In a similar way the product of any number of monomials may be found. Thus, find the product of any two of them, as before, and then in the same way multiply the product thus found by any one of the remaining monomials, and so on until all the monomials have been used; the last product will be the product required. For example; to find the product of 7ab, 9a2bc, and -5a3c, we have 7ab × 9abc × — 5a3c, = − 5 × 7 × 9a1a2 a3b1b1c1c1 — — 315abc, the sign of the product being -, because two of the monomials are positive, and the other negative. = For another example; we shall find the product of 3ab, -5ac, 6abc, and 7bd; here the product is indicated by 3ab x Бас х 6abc x 7bd 3 x 5 X = 6 × 7 × a1a1a1 b1b1b11c1d = 630abcd, the sign of the product being +, because the number of negative monomials is an even number. EXAMPLES. 1. Find the product of 2mn Vcd, - 3pqrs, -7mn3pp qqq √r, and 13xyz. 2. Find the product of 13ab3, - 17a2bc, and 9cd. 3. Find the product which is indicated by -5a × -7b X - 13ab x - 25de. (12.) We will now show how to find the product of any compound quantity whose terms are regarded as integral, by any integral monomial quantity. Let abc stand for the compound quantity, and d for the monomial; then, by the nature of multiplication, the product will be expressed by (a + b − c) + (a + b · c) + (a+b-c), etc.; until (a+b-c) is taken as many times in the sum as there are units in d. = Now we have (a + b −c) + (a + b − c) + (a + b −c) +, etc., to d terms, which gives d (a+b-c) da + db - de, or (a+bc)dad + bd cd, since, by what has been shown, the factors of each product may be interchanged. It is easy to see that if the multiplier had been d, we Hence results the following rule for the multiplication of any compound quantity by any monomial quantity. RULE. Multiply each term of the compound quantity by the monomial quantity, according to the rule for multiplying monomials, and add the products, or connect them by their signs, and the sum thus found will be the product required. Thus, to multiply 5a-3b+2c by 4ab, we shall have (5a3b+2c) x 4ab20ab12ab+8abe for the × duct. -- pro Also, the product of 3ab3+2mnp-5cde and 7ed is indicated by (3a2b3 + 2mnp - 5cde) x 7cd, which, when the multiplication is executed, gives (3a2b3 + 2mnp — - 5cde) x7cd21a2bcd14cdmnp + 35c'de. am + am-16 — am — 272 — am—373 + am - 474 (13.) Reversely, from the rules that have been given for the multiplication of a monomial, or compound quantity, by a monomial quantity, we can often resolve a given algebraic expression into factors, or find quantities such that their product shall equal the given quantity. We shall call any algebraic expression that is formed from the multiplication of other expressions that are not explicitly of fractional forms, a composite quantity; and any algebraic expression that is not thus formed, we shall denominate a simple or prime quantity. If any algebraic expression is not of a fractional form, we shall call it an integral form. We will now give a few examples, for the purpose of showing how to resolve them into their factors. For a first example, we shall take 3ab, whose prime factors are easily seen to be the primes 3, a, and b, for the product |