The first division gives at once the remainder (y-1)x+2y; taking this remainder for a divisor, we obtain, without any preparation, the remainder y2 —1. We shall obtain then all the solutions of the proposed system by resolving the equations y2-1=0, (y-1)x+2y=0. The first equation gives y=±1. For the value y=-1 we find x=—1, and this system will satisfy the proposed equations. For the value y=+1 we find r∞. This system, also, will satisfy the proposed equations; for dividing each of these equations by the highest power of x, and taking x=∞, the two equations will be reduced to y-1=0. EXAMPLE II. (y-1)x2+yx+y2-2y=0, The division gives the remainder y2-2y=0; the solutions, therefore, of the proposed equations depend on the system y2—2y=0, (y—1)x+y=0. These equations give the two systems y=0,x=0; y=2, x=−2. But the proposed equations possess, besides, another solution, y=1, x=∞, since the value y=1 causes the multiplier of the highest power of x in each of these equations to vanish. 322. The following method of elimination avoids the introduction of foreign roots, and enables us to determine the degree of the final equation: Let equation A or 2"+P+Q.....+T+V be supposed equal to C; (x-a)(x+Ax-2+Вx-3+, &c.) .... and equation B or x2+P'x"¬1+Q'.x"¬2 ... +T'x+V' to (x-a)(x+A'x"-2+B'x"-3+, &c.) . . . . D; -1 also, let equation A be multiplied by +A'"+B'-3, &c., and equation B be multiplied by x-1+AxTM-2+Brm-3, &c., it is evident that the products must be equal; therefore, m-3 ... . (xTM+ P ̧TM¬1+Q2TM−2+, &c.)(x"¬1+A'xa¬2+B'x"¬3+, &c.)=(x”+P'x"¬1+ Q'x"−2+, &c.)(.x1¬1+Ax1¬2+BxTM¬3+, &c.) . E. Performing the multiplications and making equal to each other, the coefficients of the same powers of x (Art. 209), m+n-1 equations are obtained between the indeterminate quantities A, B, C,.... A', B', C',............. Now, the number of indeterminate quantities in equation C is m-1, and in equation D, n-1; therefore, the number in equation E is m+n-2. Of the m+n−1 equations m+n-2 suffice to determine A, B, C,...A', B', C',....; and one equation remains between P, Q, R.... P', Q', R' which it is necessary to satisfy in such a manner that the equations C, D may have a common divisor, x-a; this equation of condition is the final equation required. ..... .... Since none of the indeterminate quantities A, B, C....... A', B', C' .......... is multiplied by itself, the equations by means of which these quantities are determined are of the first degree. The final equation being resolved, and the values of y successively substituted in A, B, C. . . . . A', B', C', . . ., the results are obtained from the division of the polynomials C, D by the common divisor r-a. If the equations A, B are incomplete, the two products E can not be complete polynomials of the degree m+n−1; but the terms which are deficient in the one are found in the other. For, taking the least favorable case, viz., x+P=0; x2+P'=0; the identity which results from the equality of the two products is Let (x+P)(x+A'xa¬+, &c.)=(x2+P')(x-1+Ax+, &c.) EXAMPLE. x2+Pr+Q=0; x2+P'x+Q'=0. Denoting by x-a the factor which is to be rendered common to these equations by the suitable determination of y, the first equation may be considered the product of x-a by a factor, r+A, of the first degree; and the second the product of x—a by a factor, x+A', also of the first degree. Making the coefficients of the same powers of x equal to each other, P+A' P'+A or A-A' P-P' ... ... (1) (2) (3) By mean of these three equations of the first degree the two indeterminate quantities A, A' can be eliminated, and a single equation obtained in terms of the quantities P, Q, P', Q'. For, if from equation (1), multiplied by P, or AP-PA' (P-P')P, equation (2) be subtracted, or AP'-PA'=Q-Q', the remainder is If these values of A, A' are substituted in equation (3), or or or (P—P')PQ'—(Q—Q')Q'—(P—P')QP'+(Q—Q')Q=0, The quantities P, P', Q, Q', containing only y and known quantities, this is the final equation in y. It has been already noticed that, if this equation is identical, the proposed equations have at least one common factor of the form x-a, whatever be the value of y; and that, if it contains only known quantities, these equations are contradictory. When the final equation has the proper form, the factor x-a is obtained by dividing the first of the proposed equations by x+A; thus, x+A) x2+Px+Q (x+P−A x2+Ax (P-A)x+Q (P-A)+(P.A)A Q-(P-A)A. The quotient is x+P−A, and the remainder is considered equal to zero, because it is reduced to zero by the substitution, for y, of a value deduced from the final equation. Making the quotient r+P-A equal to zero, the value of x is x=A-P, and by substituting the value of A, x= (P—P')P—(Q-Q')_P, P-P' or x=- P-P This example is given as an illustration of the general method. From its particular form it admits of resolution by another and a much shorter process. For if from the remainder is x2+Px+Q=0 x2+P'x+Q=0 is subtracted, OF THE DEGREE OF THE FINAL EQUATION. 323. The degree of the final equation can not be depressed by the reduction of each of the coefficients P, Q, R... P', Q', R'... in the equations to the term of the highest exponent in y which it contains; for the degree of each of the equations is not changed by the reduction. Therefore, the reasoning may be applied to the equations which are of the same degree respectively as the preceding equations. The latter are reducible to these equations may be decomposed into (--)(-)->). &c. =0, (-)(-)(-Y), &c. =0. -B' Substituting in equation (5) the roots of x from equation (6), viz., a'y, B'y, &c., (a'y-ay)(a'y-ẞy) (a'y-yy), &c. =0, Or, since the number of factors in equation (5) is m, and the number of roots in equation (6) is n, y(a'-a)(a-B)(a'—y), &c. =0, yTM (ẞ'—a)(ẞ' —ẞ)(B'—y), &c. =0, Consequently, there are n equations, each of the degree m; these give all the solutions in y. The product of these roots (or solutions) of y is the final equation, since it becomes zero for all the values of y which render its factors zero, and only for these values. Now, this product is evidently of the degree mn. Consequently, the degree of the final equation (unless roots not belonging to the proposed equations are introduced by the process of elimination) can not exceed the product of the degrees of the proposed equations. It ought to be observed that the numerical values of the roots of y are changed by this process, but that their number remains undisturbed by it. IRRATIONAL EQUATIONS. 324. All the direct methods employed for the solution of equations suppose that the unknown quantities in them are not affected with any radical sign; when, therefore, the unknown is found under a radical sign, it will be necessary, before applying the process of solution, to employ some preparatory method of rendering the equation rational. Such a method is at once suggested by the theory of elimination. For, if we equate each of the irrational terms with an unknown quantity, and remove the radical from each of these new equations by involution, we shall have a series of equations (including the original one, with its irrational terms replaced by the new symbols) without radicals, from which the quantities, temporarily introduced, may be eliminated, and thence a rational equation obtained, involving only the original unknown quantities. The following examples will fully illustrate the mode of proceeding: (1) Let the equation be Put x−√x−1+ Vx+1=0. y=√x−1, z=Vx+1; and we then have the three following rational equations, from which we may eliminate y and z, viz., y2=x—1, z3=x+1,x−y+z=0. From the last equation we get y=x+z, and, by substituting this value in the first, y becomes eliminated, and we have these two equations in x and z, viz., and, to eliminate z from these, we apply the process explained in the preceding articles, and thus get the final equation x-3x+8x+x+7x2-7x+2=0. (2) Let the equation be V4x+7+2√x—4—1. Putting y=V4x+7, z=√x—4, we have the system of equations y3=4x+7, z2=x—4, y+2z=1. EXPONENTIAL EQUATIONS. 325. An exponential equation is an equation in which the unknown appears in the form of an exponent or index; thus, the following are exponential equations : The value of x' lies evidently between 3 and 4; place it, therefore, equal to 3 plus an unknown fraction, and we shall have and proceed as before. The value of x is thus obtained in a continued fraction. 1 1 which may be carried to any method explained hereafter. 1 1 3+ 3+x", &c., extent at pleasure, and the value found by the (See Continued Fractions.) When the equation is of the form a2=b, or a11=c, the value of x is readily obtained by logarithms, as we have already seen in Art. 220. But if the equation be of the form ra, the value of r may be obtained by the rule of double position, as in the following EXAMPLE. Given x=100, to find an approximate value of x. * Exponential equations, and those in which logarithms of unknown quantities enter, belong to a class called transcendental. |