5x4 52 x3 201 x2 342x+216 = 0; 18. the divisor common to this and the proposed equation is x 3 8 x2 + 21x As this divisor is of the third degree, it must itself contain several factors; we must therefore seek, whether it does not contain some that are common to the equation B = O, which is here We find, in fact, for a result x3; the proposed equation then has three roots equal to 3, or admits of (x3)3 among the number of its factors. Dividing the first common divisor by x3, as many times as possible, that is, in this case twice, we obtain x As this divisor is common only to the proposed equation, and to the equation A= 0, it can enter only twice into the proposed equation. It is evident then, that this equation is equivalent to с (x-3)3(x-2)2 = 0. 208. As the equation (d) gives the difference between b and the several other roots, when b is substituted for a, the difference between c and the others, when c is substituted for a, &c. and undergoes no change in its form by these several substitutions, retaining the coefficients belonging to the equation proposed, it may be converted into a general equation, which shall give all the differences between the several roots combined two and two. For this purpose, it is only necessary to eliminate a by means of the equation am+Pam−1 + Qam-2, + Ta+U = 0; for the result being expressed simply by the coefficients, and exhibiting the root under consideration in no form whatever, answers alike to all the roots. It is evident, that the final equation must be raised to the degree m (m — 1); for its roots - d, &c. b, b a, b C- a, с c - d, &c. are equal in number to the number of arrangements, which the m letters, a, b, c, &c. admit of when taken two and two. Moreover, since the quantities a, b c and c. b, &c. differ only in the sign, the roots of the equation are equal, when taken two and two, independently of the signs; so that if we have ya, we shall have, at the same time, y=— a. Hence it follows, that this equation must be made up of terms involving only even powers of the unknown quantity; for its first member must be the product of a certain number of factors of the second degree of the form (yα) (y +α) (184); it will, therefore, itself be exhibited under the form 2n-2 y3n + py2n-2 + q y2n→ . . . . . + ty3 + u = 0. If we put y2 = 2, this becomes 2" + p zn−1 + 9 = " ..... .... · +tz + u = 0; and as the unknown quantity z is the square of y, its values will be the squares of the differences between the roots of the proposed equation. It may be observed that as the differences between the real roots of the proposed equation are necessarily real, their squares will be positive, and consequently the equation in z will have only positive roots, if the proposed equation admits of those only, which are real. Let there be, for example, the equation Suppressing the terms a37a+ 7, which, from their identity with the proposed equation, become nothing when united, and dividing the remainder by y, we have 3a2 + 3 ay + y2 — 7 = 0; eliminating a by means of this equation and the equation y42y+441 y 490; putting z = y3, this becomes 42 z+441 z — 49 0. 209. The substitution of a + y in the place of x in the equation xm + P xm−1 + Q xm-2 + U=0 (204), ... is sometimes resorted to also in order to make one of the terms of this equation to disappear. We then arrange the result with reference to the powers of y, which takes the place of the unknown quantity x, and consider a as a second unknown quan tity, which is determined by putting equal to zero the coefficient of the term we wish to cancel; in this way we obtain If the term we would suppress be the second, or that which involves y, we make ma + P= 0, from which we deduce Hence it follows, that we make the second term of an equation to disappear, by substituting for the unknown quantity in this equation a new unknown quantity, united with the coefficient of the second term taken with the sign contrary to that originally belonging to it, and divided by the exponent of the first term. Let there be, for example, the equation x36x-3x+4=0; in which the term involving y2 does not appear. We may cause the third term, or that involving y-2, to disappear by putting equal to zero the sum of the quantities, by which it is multiplied, that is, by forming the equation m (m 1 2 Pursuing this method, we shall readily perceive, that the fourth term will be made to vanish by means of an equation of the third degree, and so on to the last, which can be made to disappear only by means of the equation am + Pam-1 + Qam-2 + U = 0, perfectly similar to the equation proposed. It is not difficult to discover the reason of this similarity. By making the last term of the equation in y equal to zero, we suppose, that one of the values of this unknown quantity is zero; and if we admit this supposition with respect to the equation x = y + a, it follows that x = a; that is, the quantity a, in this case, is necessarily one of the values of x. 210. We have sometimes occasion to resolve equations into factors of the second and higher degrees. I cannot here explain in detail the several processes, which may be employed for this purpose; one example only will be given. Let there be the equation in which it is required to determine the factors of the third degree; I shall represent one of these factors by x2 + p x2 + q x + r, the coefficients, p, q, and r, being indeterminate. They must be such, that the first member of the proposed equation will be exactly divisible by the factor x2 + px2 + qx+r, independently of any particular value of a; but in making an actual division, we meet with a remainder an expression, which must be reduced to nothing, independently of x, when we substitute for the letters, p, q, and r, the values that answer to the conditions of the question. We have then p32pq- 24p+r12 = 0, These three equations furnish us with the means of determining the unknown quantities, p, q, and r; and it is to a resolution of these, that the proposed question is reduced. Of the Resolution of Numerical Equations by Approximation. 211. HAVING Completed the investigation of commensurable divisors, we must have recourse to the methods of finding roots by approximation, which depend on the following principle; When we arrive at two quantities which, substituted in the place of the unknown quantity in an equation, lead to two results with contrary signs, we may infer, that one of the roots of the proposed equation lies between these two quantities, and is consequently real. Let there be, for example, the equation if we substitute, successively, 2 and 20 in the place of x, in the first member, instead of being reduced to zero, this member becomes, in the former case, equal to 31, and in the latter, to +2939; we may therefore conclude, that this equation has a real root between 2 and 20, that is, greater than two and less than 20. As there will be frequent occasion to express this relation, I shall employ the signs > and <, which algebraists have adopted to denote the inequality of two magnitudes, placing the greater of two quantities opposite the opening of the lines, and the less against the point of meeting. Thus I shall write x2, to denote, that is greater than 2, x20, to denote, that x is less than 20. Now in order to prove what has been laid down above, we may reason in the following manner. Bringing together the positive terms of the proposed equation, and also those which are negative, we have x37x-(13x2 + 1), a quantity, which will be negative, if we suppose x = 2, because, upon this supposition, x37x13x2 + 1, and which becomes positive, when we make x = 20, because, in this case, x37x13x2 + 1. Moreover, it is evident, that the quantities x2+7x each increase, as greater and greater values are assigned to x, and that, by taking values, which approach each other very nearly, we may make the increments of the proposed quantities as small as we please. But since the first of the above quantities, which was originally less than the second, becomes greater, it is evident, that it increases more rapidly than the other, in consequence of which its deficiency is made up, and it comes at length |
