The same is true of literal expressions when all the terms are connected together by the sign +. and is obtained by multiplying each part of the multiplicand by the multiplier, and adding together the two particular products ac and bc. The operation is the same when the multiplicand contains more than two parts. If the multiplier is composed of several terms, it is manifest that the product is made up of the sum of the products of the multiplicand by each term of the multiplier. The product of multiplied by a+b c + d for by multiplying first a + b by c, we obtain a c + bc, then by multiplying a + b by the second term d of the multiplier, we have a d+bd, and the sum of the two results gives for the whole. ac+be+ad + bd 29. When the multiplicand contains parts to be subtracted, the products of these parts by the multiplier must be taken from the others, or in other words, have the sign prefixed to them. for each time that we take the entire quantity a, which was to have been diminished by b before the multiplication, we take the quantity b too much; the product a c therefore, in which the whole of a is taken as many times as is denoted by the number o, exceeds the product sought by the quantity b, taken as many times as is denoted by the number c, that is by the product bc; we ought then to subtract be from a c, which gives, as above, ac- bc. The same reasoning will apply to each of the parts of the mul tiplicand, that are to be subtracted, whatever may be the num ber, and whatever may be that of the terms of the multiplier, pro vided they all have the sign +. Recollecting that the terms which have no sign are considered as having the sign +, we see by the examples, that the terms of the multiplicand affected by the sign give a product affected by the sign +, while those which have the sign — give one having the sign It follows from this, that when the multiplier has the sign +, the product has the same sign as the corresponding part of the multiplicand. 30. The contrary takes place when the multiplier contains parts to be subtracted; the products arising from these parts must be put down with a sign, contrary to that which they would have had by the above rule. This may be shown by the following example. for the product of the multiplicand, by the first term of the multiplier, will be by the last example a cbc; but by taking the whole of c for the multiplier instead of c diminished by d, we take the quantity ab so many times too much as is denoted by the number d; so that the product ac-bc exceeds that sought by the product of ab by d. Now this last is, by what has been said, a db d, and in order to subtract it from the first it is necessary to change the signs (20). We have then a c bc adbd for the result required. 31. Agreeably to the above examples, we conclude, that the multiplication of polynomials is performed by multiplying successively, according to the rules given for simple quantities (21—26), all the terms of the multiplicand by each term of the multiplier, and by observing that each particular product must have the same sign, as the corresponding part of the multiplicand, when the multiplier has the sign +, and the contrary sign when the individual multiplier has the sign If we develop the different cases of this last rule, we shall find, 1. That a term having the sign +, multiplied by a term having the sign+, gives a product having the sign +; 2. That a term having the sign, multiplied by a term having the sign+, gives a product which has the sign; 3. That a term having the sign +, multiplied by a term having the sign, gives a product which has the sign —; 4. That a term having the sign, multiplied by a term having the sign, gives a product which has the sign +. -It is evident from this table, that when the multiplicand and multiplier have the same sign, the product has the sign +, and that when they have different signs, the product has the sign To facilitate the practice of the multiplication of polynomials, I have subjoined a recapitulation of the rules to be observed. 1. To determine the sign of each particular product according to the rule just given; this is the rule for the signs. 2. To form the coefficients by taking the product of those of each multiplicand and multiplier (22); this is the rule for the coefficients. 3. To write in order, one after the other, the different letters contained in each multiplicand and multiplier (21); this is the rule for the letters. 4. To give to the letters, common to the multiplicand and multiplier, an exponent equal to the sum of the exponents of these letters in the multiplicand and multiplier (25); this is the rule for the exponents. 32. The example below will illustrate all these rules. Result reduced 5a7-22ab+12a5b2-6a4b3-4a3b1+8a3b5. The first line of the several products contains those of all the terms of the multiplicand by the first term a3 of the multiplier; this term being considered as having the sign +, the products which it gives have the same signs as the corresponding terms of the multiplicand (31). The first term 5 a of the multiplicand having the sign plus, we do not write that of the first term of the product, which would be; the coefficient 5 of a being multiplied by the coefficient 1 of a3, gives 5 for the coefficient of this product; the sum of the two exponents of the letter a is 4 + 3, or 7, the first term of the product then is 5 a7. The second term 2 a3 b of the multiplicand having the sign -, the product has the sign minus; the coefficient 2 of a b mul tiplied by the coefficient 1 of a3, gives 2 for the coefficient of the product; the exponent of the letter a, common to the two terms which we multiply, is 3 + 3, or 6, and we write after it the letter b, which is found only in the multiplicand. The second term of the product then is 2 a® b. α The third term + 4 a2 ba gives a product affected with the sign+, and by the rules applied to the two preceding terms, we find it to be + 4 a5 b2. The second line contains the products of all the terms of the multiplicand by the second term - 4a3 b of the multiplier. This last having the sign, all the products which it gives must have the signs contrary to those of the corresponding terms of the multiplicand; the coefficients, the letters, and the exponents are determined as in the preceding line. The third line contains the products of all the terms of the multiplicand by the third term +263 of the multiplier. This term having the sign +, all the products which it gives have the same sign as the corresponding terms of the multiplicand. After having formed all the several products which compose the whole product, we examine carefully this last, to see whether it does not contain similar terms; if it does, we reduce them according to the rule (19), observing that two terms are similar, which consist of the same letters under the same exponents. In this example there are three reductions, viz; 2 a b and +4 a5 b2 and + 20 a b, 22 a® b; 6 a4 b3. These reductions being made, we have for the result the last line of the example. See another example to exercise the learner, which is easily performed after what has been said. Multiplicand 5a1b2+7a3b3—15a5c+23b2 d1—17bc3 d2—9abcdm3 55a4b5+77a3b6-165a5b3c+253b5d4-187b4c3d2-99ab*cdm3 5 +120a5 c1-184b2c3d+136bcd2+72abcdm2+35ab4c-75abe2+115ab3cd1-85ab2c+d= -45a2b2 c2 dm3—10a1b3 dm-14a3b dm+30a5bcdm-46b3d3m+34b3c3d3m+18ab2cd3m3. |