The third line is formed by adding to the numbers found in the preceding the coefficient - - 20, by which x is multiplied; this is The fourth line contains the quotients of the several numbers in the preceding, divided by the corresponding divisors; this is the R a line for the quantities In forming this line, we neglect all the numbers, which are not entire. The fifth line results from the numbers, written in the preceding, added to the number 23, by which 2 is multiplied; this line contains the quantities Q'. The sixth line contains the quotients arising from the numbers in the preceding, divided by the corresponding divisors; it compre hends the quantities Q' a The seventh line comprehends the several sums of the numbers in the preceding, added to the coefficient 9, by which a3 is mul Q' tiplied; in this line are found the quantities. + P. a Lastly, the eighth line is formed, by dividing the several numbers in the preceding by the corresponding divisors; it is the line for As we find 1 only in the column, at the head of which +3 stands, we conclude, that the proposed equation has only one commensurable root, namely +3; it is, therefore, divisible by x-3.* The divisors 1 and 1 may be omitted in the table, as it is easier to make trial of them, by substituting them immediately in the proposed equation. Having ascertained, that the numbers + 1 and 1 do not satisfy this equation, we form the table subjoined, according to the preceding rules, observing that, as the term involving x is wanting in this equation, a must be regarded as having 0 for a coefficient; we must, therefore, suppress the third line, and deduce the fourth immediately from the second. *Forming the quotient according to the preceding note, we find x3- 6 x2 + 5 x — 5. - +36, +18, +12, + 9, +6, + 4, + 3, + 2, 2, 3, 4, 6, — 9, — 12, — 18,- 36 + 1, 2, + 3, + 4, + 6, † 9, + 12, † 18, — 18, — 12, — 9, — 6, — 4, — 3, — 2, — 1 ************************************************************************** + 1, + 4, + 9, + 9, + 4, + 1, same time, the three roots, which the proposed equation admits of; we conclude then, that it is the product of three simple factors, Ꮖ 6, x 3, and x + 2. 203. It may be observed, that there are literal equations, which may be transformed, at once, into numerical ones. If we have, for example, y3+2py2-33 p2 y + 14 p30, As the commensurable divisor of this last equation is x + 7, which gives x = 7, we have y=-7p. The equation involving y is among those which are called homogeneous equations, because taken independently of the numerical coefficients, the several terms contain the same number of factors.* 204. When we have determined one of the roots of an equation, we may take for an unknown quantity the difference between this root and any one of the others; by this means we arrive at an equation of a degree inferior to that of the equation proposed, and which presents several remarkable properties. Let there be the general equation x+Pxm-1+Q xm−2+ R xm−3 ..... Tx+U=0. and let a, b, c, d, &c. be its roots; substituting ay in the place of x, and developing the powers, we have * For a more full account of the commensurable divisors of equations, the reader is referred to the third part of the Elémens d'Algébre of Clairaut. This geometer has treated of literal as well as numerical equations. +Qam−2+(m—2) Qam—3y+ (m—2)(m— -3) Qam-4y2+... 2 (m—3)(m—1) Ram—5y2+... =0. The first column of this result, being similar to the proposed equation, vanishes of itself, since a is one of the roots of this equation; we may, therefore, suppress this column, and divide all the remaining terms by y; the equation then becomes mam-1 + m(m-1) am-2y +...+ym-1 +T=A. = B, abridging the expressions, by making m am−1 + (m − 1) P am−2 + (m −2) Q am and I shall designate by the expression am+Pam-1 + Qam-2 ..... + Ta+U. 205. If the proposed equation has two equal roots; if we have, for example ab, one of the values of y, namely ba, becomes nothing; the equation (d) will therefore be verified, by supposing y = 0; but upon this supposition all the terms vanish, except the known term A; this last must, therefore, be nothing of itself; the value of a must, therefore, satisfy, at the same time, the two equations V0 and A = 0. When the proposed equation has three roots equal to a, namely, a = b = c, two of the roots of the equation (d) become nothing, at the same time, namely, b- a and c -α. In this case the equation (d) will be divisible twice successively by y―0 (179) or y; but this can happen, only when the coefficients A and B are nothing; the value of a must then satisfy, at the same time, the three equations V = 0, 0, B = 0. Pursuing the same reasoning, we shall perceive, that when the proposed equation has four equal roots, the equation (d) will have three roots equal to zero, or will be divisible three times successively by y; the coefficients, A, B, and C, must then be nothing, at the same time, and consequently the value of a must satisfy at once the four equations, V = 0, A = 0, B = 0, C = 0. By means of what has been said, we shall not only be able to ascertain, whether a given root is found several times among the roots of the proposed equation, but may deduce a method of determining, whether this equation has roots repeated, of which we are ignorant. or For this purpose, it may be observed, that when we have A = 0, mam-1+(m—1) Pam-2+ (m-2) Qam-3... + T = 0, we may consider a as the root of the equation mxm-1+ (m − 1) Pxm2 + (m-2) Qxm-3... + T = 0, x representing, in this case, any unknown quantity whatever; and since a is also the root of the equation = 0, or xm + Pxm−1 + &c. = 0, it follows (189), that x a is a factor common to the two above mentioned equations. Changing in the same manner a into x in the quantities, B, C, &c. the binomial x a becomes likewise a factor of the two new equations, B = 0, C= 0, &c. if the root a reduces to nothing the original quantities, B, C, &c. What has been said with respect to the root a, may be applied to every other root, which is several times repeated; thus, by seeking, Alg. 29 according to the method given for finding the greatest common divisor, the factors common to the equations, V=0, A = 0, B = 0, C = 0, &c. we shall be furnished with the equal roots of the proposed equation, in the following order; = α) The factors common to the first two equations only, are twice factors in the equation proposed; that is, if we find for a common divisor of V = 0 and A = 0, an expression of the form (x (x 6), for example, the unknown quantity x will have two values equal to a, and two equal to 6, or the proposed equation will have these four factors, (x — α), (x — α), (x — 6), ( 6). The factors common, at the same time, to the first three of the above mentioned equations form triple factors in the proposed equation; that is, if the former are presented under the form (x — α) (x — 6), the latter will take the form, (x — a)3 (α — 6)3. This reasoning may easily be extended to any length we please. 206. It may be remarked, that the equation A= 0, which, by changing a into x, becomes mxm-1+(m—1) Pxm2 + (m −2) Q xm-3..+T= 0, is deduced immediately from the equation = 0, or from the proposed equation, V xm+Pxm-1+Q xm-2... + Tx + U = 0, by multiplying each term of this last by the exponent of the power of x, which it contains, and then diminishing this exponent by unity. We may remark here, that the term U, which is equivalent to UX x, is reduced to nothing in this operation, where it is multiplied by 0. The equation B = 0 is obtained from A = 0, in the same manner as A = 0 is deduced from = 0; C = 0 is obtained from B = O, in the same manner as this from A = 0, and so on.* *It is shown, though very imperfectly, in most elementary treatises, that the divisor common to the two equations V0 and A0 contains equal factors raised to a power less by unity than that of the equation proposed; this may be readily inferred from what precedes; but for a demonstration of this proposition we refer the reader to the Supplement, where it is proved in a manner, which appears to me to be simple and new. |