CHAPTER VIII. ON THE FORMATION OF BINOMIAL PRODUCTS AND POWERS. other, laws which are found in the of binomial and 477. THE products of binomial and other factors, whose Important terms possess no peculiar or assigned relation to each may be easily formed, in all cases, by the general rule which to prevail is given for that purpose (Art. 61): but there are some laws formation which are found to prevail in the formation of the products products of binomial factors which have the same first term, and more particularly in the products or powers of binomial factors which are identical with each other, which not only furnish most rapid and convenient rules for the performance of the operation of multiplication and involution in such cases, but either constitute or lead to, the most important theorems both in Arithmetical and Symbolical Algebra. powers. 478. We will begin with the formation of the product of Products of two binotwo binomial factors xa and x+b*, which have the same mial factors first term a the process exhibited at full length, will stand which have as follows: the same first term. x + a x + b x2 + ax + bx+ab x2 + (a + b)x + ab The terms of this product are arranged according to the powers of the letter a, and consequently the terms ax and ba, both involving the simple power of x, are like terms (Art. 28), See Examples 1, 2, 3, 4, Art. 69. Product of three bino and are therefore, when added together, collected into the single term (a+b)x: it consequently appears that the coefficient of x, in the second term, is the sum of the second terms of the binomial factors xa and x + b, and that the last term is their product*. 479. Let it now be required to form the product of three mial factors binomial factors x+a, x+b and x+c, which have the same having the first term : if we multiply the product same first term. x2 + (a + b) x + ab of the two first factors x+a and a+b (Art. 478.) into the third factor x + c, the process will stand as follows: x2 + (a + b)x + ab x + c a13 + (a + b) x2 + abx + cx2 + (a + b) cx + abc x3 + (a + b + c) x2 + (ab + ac + bc)x + abc. This result is arranged according to the powers of the letter r, and consequently (a+b)x2 and ca2, being like terms, make, when added together, the single term (a+b+c) x2+: in a similar manner, the like terms abx and (a+b)cx, make, when added together, the single term (ab+ ac+bc)x it appears therefore that the coefficient (a+b+c) of the second term of this product is the sum of the second terms of the several binomial factors: that the coefficient of the third term or ab + ac + bc, is the sum of all their products two and two, and that the last term is their continued product. *The product of a and b will be found to be 2-(a + b) 1 + ab, differing from the former in the sign of the second term only, which is negative the product of x+a and x-h is x2+ (a - b) x − ab, and that of - a and b is a2 - (a - b) x — ab. : For the sum of their coefficients, (a+b) and c, is a+b+c. For the sum of their coefficients ab and (a+b)c, or ab and ac + be (for (a + b) c = ac + bc, Art. 45) is ab+ ac + be: by putting this sum under the form ab + ac + be instead of ab + (a + b) c, we make its symmetrical composition, in terms of a, and c, not only more manifest to the eye, but also more easily expressible in ordinary language. four bino 480. If we should multiply the product given in the last Product of Article by an additional factor x+d, we should find for the mial factors result x* + (a + b + c + d) x3 + (ab + ac + ad + b c + bd + cd) x2 which is arranged according to the descending powers of x, and having the same first term. formation 481. The law of formation of the products of such binomial The law of factors is now sufficiently manifest, and we are authorized there- of such products fore to assume hypothetically that the same law will prevail in the generalformation of the product of (a) such factors: and it may be ized. easily demonstrated that if it be true for (n-1) such factors, it must necessarily be true for a number of them expressed by the next superior number n. (Art. 447.) 482. For let us suppose the (n-1) first factors to be ex- Its proof. pressed by x + a1, X + A2, . . . X + A,-*, and their product by where the coefficients P1, P2, P3,... of the descending powers of x, are respectively the sum of all the second terms of the several binomial factors, the sum of all their products two and twot, the sum of all their products three and three, and so on, the last term p being their continued product. If we now suppose this assumed product of (n-1) binomial factors * In denoting a series of quantities, where both their number and order of succession are required to be expressed, we adopt this notation, in preference to ra, x + b, x + c, x+d, &c. where the successive letters of the alphabet are employed to denote the successive second terms of the binomial factors. By all the products two and two, we mean all those which can be formed by every possible combination of the letters a, ag, and a-1, taken two and two together: the same remark applies to the products three and three together, four and four together, &c. to be multiplied by an additional factor x+a, the process will stand as follows: representing p1+a, by q1, P2 + P1a, by q, and so on: it remains to determine the law of composition of the coefficients of the final product in terms of a1, ɑ„,...ɑ„. or is equal to the sum of the second terms of all the binomial factors, since p, is, by the hypothesis, the sum of the second terms of all the (n - 1) binomial factors in the first product. Again, where p is, by hypothesis, the sum of all the products, two and two which can be formed of the first (n - 1) quantities a1, a,,...ɑ-1, and where p is their sum: therefore p1a, is the sum of all the additional products, two and two, which can be formed by the new quantity a, combined with all the others a1, a,...a it follows therefore that q is the sum of all the products two and two which can be formed by the second terms of the n binomial factors, or by the n quantities a,, a,,... An Again, where P3 is the sum of all the products three and three, and P. the sum of all the products two and two, which can be formed of the (n-1) letters a, a,,... and a consequently Pa, will express the sum of all the products three and three which can be formed of the n letters a,, a,,...a,, in which a, appears as a factor: and therefore q, must necessarily express the sum of all the products three and three which can be formed of the n letters a1, a,...a„. It is obvious that the same reasoning may be applied, in the same manner, to the succeeding coefficients q, q,... qn, so that the law which has been assumed to prevail for (n − 1) binomial factors, having the same first term, must prevail likewise when the number of such factors is increased by unity but this law has been proved to prevail, when the number of factors is 2, 3 and 4: it is therefore necessarily true for 5 and if for 5, it is necessarily true for 6, and so on successively for any number of factors whatever. - When the second 483. If the binomial factors were rα1, x − α, x — x-a, whose second terms a1, α, are severally sub- terms of ... an tracted from the first, their product would be expressed by where the coefficients 91, 92, 93, &c. are the same as in the last Article, but the terms are alternately negative and positive, beginning with the second. 484. The following are examples: (1) (x+3)(x+5)= x2+8x+15, where 8 = 3 +5 and 15 = 3 × 5...(Art. 478). ... would If it was required to express the product of a series of mixed binomial factors, such as r+a1, r-αg, x-αz, x+a4, &c. where some of the second terms are positive and others negative, the coefficients q1, 42, severally express the same series of combinations of the letters a1, a,... a,, but those combinations would be negative or positive according as an odd or an even number of letters, preceded by negative signs, entered into each product: thus, the product of x+a, I – b and r+c, would be x3 + ( a − b + c ) x2 – (ab − a c + be) x — abc: the product of x + a, x − b and xc, would be x3 + ( a − b −c) x2 - (ab+ ac-bc)x + abc: the product of r -a, x-b, x+c and x-d, would be x1 − (a + b − c + d) x3 + ( a b − ac + ad− b c + bd - cd) r2 +(abe - abd + aed + bed) x - abcd: the formation however of such products, as included in the general theorem, given in Article 482, will be easily understood in Symbolical Algebra, where symbols are considered, at pleasure, as negative or positive, per se, and not in virtue merely of the sign of the operation, whether of subtraction or addition, which precedes them. the bino mial fac tors are negative. 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