SECTION VIII. PROPERTIES OF EQUATIONS. m 419. Let us assume the equation, +Ax*-+B.cm-'+...+ Tx+U= 0, (1) in which m, the exponent of the degree, is a positive whole number. An equation not given in this form may be readily reduced to it, by transposing all the terms to the first member, arranging them according to the descending powers of the unknown quantity, and dividing through by the coefficient of the first term. In this equation the coefficients, A, B, C, etc., may denote any quantities whatever; that is, they may be positive or negative, entire or fractional, rational or irrational, real or imaginary. The term U may be regarded as the coefficient of xo, and is called the absolute term of the equation. 420. If the equation contains all the entire powers of X, from the mth down to the zero power, it is said to be complete; if some of the intermediate powers of x are wanting, it is said to be incomplete. An incomplete equation may be made to take the form of a complete equation, by writing the absent powers of x with +0 for their coefficient. 421. It has been shown (305) that any expression of the second degree whatever, containing but one unknown quantity, may be resolved into two binomial factors of the first degree with respect to the unknown quantity,—the first term in each factor being this quantity, and the second term one of the roots (with its sign changed) of the equation which results from placing the expression equal to zero. We therefore conclude that every expression of the second degree may be regarded as the product of two binomial factors of the first degree. So likewise the product of three binomial factors of the first degree with respect to any unknown quantity, will be an expression of the third degree, and we readily see that by varying the values of the second terms of the factors, corresponding changes are produced in the product. Thus, (x-2)(x+3)(x–5)=x'—4x*—11x+30, and, (2–2+1=3)(2–2–V=3)(x+ 1) = x*— fx9+; also, (x+1-V-3)(x+1+1=3)(x-2) = 2°—8. From these and other examples, which may be increased at pleasure, it is inferred that any expression whatever of the third degree would result from the multiplication of some three factors of the first degree in respect to X. And in general, any expression of the mth degree with respect to its unknown quantity, may be regarded as the result of the multiplication of m binomial factors of the first degree with respect to that unknown quantity. 422. If then we have any equation formed by placing a polynomial containing the unknown quantity, x, equal to zero, and we discover the binomial factor x—a in the first member, it is evident that a is a root of the equation; for, when substituted for x, it reduces the first member to zero. If we can succeed, therefore, in discovering the binomial factors of the first degree, of the first member of any equation, the roots of the equation will be the values of x obtained by placing each of these factors, successively, equal to zero. This reverse process of resolving the first member of an equation into its binomial factors of the first degree, is one the difficulty of which increases rapidly with the degree of the equation; and algebraists have as yet discovered no general method for effecting this resolution for those of a higher degree than the fourth. By special processes, however, the roots of numerical equations may be found exactly, when commensurable, and to any degree of approximation when not commensurable. 423. In order to discover the law which governs the product of any number of binomial factors, such as x+a, x+b, x+c, etc., having the first term the same in all, and the second terms different, let us first obtain the product of several of these factors by actual multiplication ; thus, 3 ta } = (x+a) (x+5) +6! oc tc cova|zo+abc+abc +6+ac! = (+a) (x+6) (+) +c+bc ac +d w*+ala+ab 12c*+abc 12+abcd + + ac tabd + + bc + acd +d fad +bcd = (x+a) (x+b) (x+c) (x+d) +od +cd From an examination of these several products we arrive at the following conclusions : 1.—The exponent of the leading letter, x, in the first term is equal to the number of binomial factors used, and this exponent decreases by 1, from term to term, towards the right, until we come to the last term, in which it is 0. 2.—The coefficient of the first term is 1; that of the second, the sum of the second terms of the binomial factors; that of the third, the sum of all the different products formed by multiplying, two and two, the second terms of the binomial factors; that of the fourth, the sum of all the different products formed by multiplying, three and three, the second terms of the binomial factors ; the last, or absolute term, is the continued product of the second terms of the binomial factors. It might be inferred from what has been now shown, that however great the number of binomial factors employed, the coefficient of that term of the arranged product which has n terms before it, would be the sum of all the different products that can be formed by multiplying the second terms of the binomial factors in sets of n and n. Assuming that the above law holds true for a number m, of binomial factors, if it can be proved that it still governs the product when an additional factor is introduced, it will be established in all its generality. Let us suppose then, that in the product, 2m+Ax-1+B1m-?+.... Mcm-n+1+Nem-"+.... + Tx+U, of the m binomial factors x+a, x+b,...., x-tp, the law of formation is the same as that found by the actual multiplication of several factors. Introducing the factor x+q, we have "+Ax-1+Bocm-2 + ....M.cm-n+1+Ncmnt.... Tx+U x+9 amti +Ax*+Bluem-it....+am-nti to...tux + Mall + Tel +Uq It is at once seen that, in this new product, the law in respect to the exponents is unbroken. As to the coefficients, that of the first term is still 1; that of the second term is A+q; and since A is the sum of the second terms of the m factors in the assumed product, A+q is the sum of the second terms of the m+1 binomials. The coefficient of the third term is B+ Aq. Now B is, by hypothesis, , the sum of the different products of the second terms, of the m binomial factors, taken two and two in a set, and Aq is all of the additional products to which the introduction of the factor x+q can give rise; hence B+ Aq is the sum of all the products, taken two and two, of the second terms of the m+1 binomials. And the coefficient of the general term, that is the coefficient of the term having n terms before it, is N+Mq; but N is the sum of all the products of the second terms, taken n and n, of the binomial factors which enter the assumed product; and because M is the sum of all the products of these second terms, taken n-1 and n-1, Mq is the sum of all the additional products, taken n and n, which can result from the introduction of the factor x+a. Now we have proved, by actual multiplication, that the law of the product, admitted to be true for n binomial factors, is true for four factors; hence by what has just been demonstrated, it is true for five factors; and being true for five, it must be true for six, and so Therefore the law is general. on. in which a, 424. The composition of the coefficients of an equation in terms of its roots. Let us take any number, m, of binomial factors, as x—a, x—6, X-C,.... 2P, X -9, b, c, etc., may represent any quantities whatever. Now it has been shown (423) that the continued product of these factors, arranged according to the desending powers of x, will be of the form, x + Axm-1 + Bxm-s+ Cam87.... Sw+To+U, in which A=-ab-CM,... -p-9, ..:} }={ (2) S= +abcd...P97–2+bcde...P9m2 + etc., U= {abcd....Pqm, the subscript expressions m—2, m_1, m, denoting the number of literal factors which enter each term. We thus have the identical equation, (1) (x-2)(C-) +Sx*+Tx+U and placing the second member of this equal to zero we have x"+Ax*-1 +B2"-2+....Sc* + Tc+U= 0 an equation of which a, b, c,.... p, q are the roots, since these values substituted in succession for x in the first member of the eq. (1) will cause this first member, and consequently the second member, to vanish. The relations between the coefficients A, B, C, etc., and the roots of eq. (2), may be expressed as follows: 1.—The coefficient of the second term is equal to the algebraic sum of all the roots, with the signs changed. 2.-The coefficient of the third term is equal to the algebraic sum of all the different products formed by multiplying the roots, two and two. 3.—The cocfficient of the fourth term is equal to the algebraic sum of all the different products formed by multiplying the roots, with their signs changed, three and three. |