Ex. 7. Find the four roots of the equation x4-55x2-30x+504-0. Ex. 8. Find the four roots of the equation x1 — 25x2+60x—36=0. Ex. 9. Find the four roots of the equation x4x3-x2+19x-42-0. Ex. 10. Find the five roots of the equation x2+5x+x3-16x2-20x-16=0.7 465. Incommensurable Roots.-If a high numerical equation is found to contain no commensurable roots, or, if after removing the commensurable roots, the depressed equation is still of a higher degree than the second, we must proceed by approximation to find the incommensurable roots. Different methods may be employed for this purpose; but the following method, which is substantially the same as published by Horner in 1819, is generally to be preferred. Find, by Sturm's Theorem, or by trial, Art. 446, the integral part of a root, and transform the given equation into another whose roots shall be less than those of the preceding by the number just found, Art. 444. Find, by Art. 446, the first figure of the root of this equation, which will be the first decimal figure of the root of the original equation. Transform the last equation into another whose roots shall be less than those of the preceding by the figure last found. Find, as before, the first figure of the root of this equation, which will be the second decimal figure of the root of the original equation. By proceeding in this manner from one transformation to another, we may discover the successive figures of the root, and may carry the approximation to any degree of accuracy required. Ex. 1. Find an approximate root of the equation x3-3x2+5x=178. We have found, page 330, that this equation has but one real root, and that it lies between 4 and 5. The first figure of the root therefore is 4. Transform this equation into another whose roots shall be less than those of the proposed equation by 4, which is done by substituting y+4 for x. We thus obtain y3+15y2+77y=46. The first figure of the root of this equation is .5. Transform the last equation into another whose roots shall be less by .5, which is done by substituting z+.5 for y. We thus obtain ≈3+16.5≈2+92.752=3.625. The first figure of the root of this equation is .03. Transform the last equation into another whose roots shall be less by .03, which is done by substituting v+.03 for z. We thus obtain v3+16.59v2+93.7427v .827623. = The first figure of the root of this equation is .008. Transform the last equation into another whose roots shall be less by .008, and thus proceed for any number of figures required. 466. How the Operation may be abridged. This method would be very tedious if we were obliged to deduce the successive equations from each other by the ordinary method of substitution; but they may be derived from each other by a simple law. Thus, let (1.) be any cubic equation, and let the first figure of its root be denoted by r, the second by r', the third by r", and so on. If we substitute r for x in equation (1), we shall have A3+Br2+Cr=V, nearly. Whence r= V (2.) If we put y for the sum of all the figures of the root except the first, we shall have x=r+y; and, substituting this value for x in equation (1), we obtain Ar3+3Ar3y+3Ary2+Ay3 + Br2+2Bry +By2 (=V; or, arranging according to the powers of y, we have Ay3+(B+3Ar) y2+(C+2Br+3Ar2)y=V—Cr−Br2 — Ar3. Let us put B' for the coefficient of y2, C' for the coefficient of y, and V' for the right member of the equation, and we have Ay3+B'y2+C'y=V'. (3.) This equation is of the same form as equation (1); and, pro ceeding in the same manner, we shall find where r' is the first figure of the root of equation (3), or the second figure of the root of equation (1). Putting for the sum of all the remaining figures, we have y=r'+z; and, substituting this value in equation (3), we shall obtain a new equation of the same form, which may be written Az3+B"z2+C"%=V"; (5.) and in the same manner we may proceed with the remaining figures. Equation (2) furnishes the value of the first figure of the root; equation (4) the second figure, and similar equations would furnish the remaining figures. Each of these expressions involves the unknown quantity which is sought, and might therefore appear to be useless in practice. When, however, the root has been found to several decimal places, the value of the terms Br and Ar2 will be very small compared with C, and r will be very nearly equal to V We may therefore employ C as an C approximate divisor, which will probably furnish a new figure of the root. Thus, in the last example, all the figures of the root after the first are found by division. 46÷77 =.5, 3.62 92.75 .03, If we multiply the first coefficient A by r, the first figure of the root, and add the product to the second coefficient, we shall have If we multiply expression (6) by r, and add the product to the third coefficient, we shall have If we multiply expression (7) by ", and subtract the product from V, we shall have V–Cr–Br2 — Aμ3 which is the quantity represented by V' in equation (3). If we multiply the first coefficient A by r, and add the product to expression (6), we shall have If we multiply expression (8) by r, and add the product to expression (7), we shall have C+2Br+3Ar2, which is the coefficient of y in equation (3). If we multiply the first coefficient A by r, and add the product to expression (8), we shall have B+3Ar, which is the coefficient of y2 in equation (3). We have thus obtained the coefficients of the first transformeà equation; and, by operating in the same manner upon these coefficients, we shall obtain the coefficients of the second transformed equation, and so on; and the successive figures of the root are indicated by dividing V by C, V' by C', V" by C", etc. 467. The results of the preceding discussion are expressed in the following RULE. Represent the coefficients of the different terms by A, B, C, and the right-hand member of the equation by V. Having found r, the first figure of the root, multiply ▲ by r, and add the product to B. Set down the sum under B; multiply this sum by r, and add the product to C. Set down the sum under C; multiply it by r, and subtract the product from V; the remainder will be the FIRST DIV IDEND. 5. ་ Again, multiply A by r, and add the product to the last number under B. Multiply this sum by r, and add the product to the last number under C; this result will be the FIRST TRIAL DIVISOR. Again, multiply A by r, and add the product to the last number. under B. Find the second figure of the root by dividing the first dividend by the first trial divisor, and proceed with this second figure precisely as was done with the first figure, carefully regarding the local value of the figures. The second figure of the root obtained by division will fre qently furnish a result too large to be subtracted from the remainder V', in which case we must assume a different figure. After the second figure of the root has been obtained, there will seldom be any further uncertainty of this kind. It may happen that one of the trial divisors becomes zero. In this case equation (2) becomes that is, the next figure of the root will be indicated by dividing the last dividend by the last number under B, and extracting the square root of the quotient. The entire operation for finding a root of the equation Having found one root, we may depress the equation. x3-3x2+5x-178-0 |