stance, a little examination will show that the equation is decomposable into the two quadratics x+4x+2=0, and x-4x-3=0. 95. It may be further observed that in determining a superior limit to the positive roots of an equation, as above, we should always so manage the decomposition into factors as to secure under the radical the greatest negative quantity we can. Thus if x+x-13=0 were proposed, then changing it into x+x-13x=0, we might decompose thus: +√(x2+ 13x), which suggests 13 for the superior limit of the positive roots; but by decomposing thus, x±√(−x1+13x), we see that 3, a much smaller superior limit, is indicated: the first figure of the positive root is in fact 2. Again, suppose the equation were x-2x+3x-20-0: if we decompose as follows, namely, x2±√(2x3—3x+20), we get no information as to the place of a positive root, though we may conclude that there can be no negative root so great as-3; but if we multiply the equation by 3x, changing it into 3x2-6x1+9x2—60x=0, and decompose thus: 3x±√(-3x+6x1+60x), 3x-10(-3x+6x+100), we learn that no positive root can be so great as 3, and thence find that the first figure of the root is 2. When the least value that renders the quantity under the radical negative is thus found, a value, still less, may always be found as follows: substitute the first value for x in the rational part, and annex the numerical result, with changed sign, to that part: the accompanying irrational part, for the same value of x, will then be a negative number greater than before. And, by repeating this process, accurate information may be obtained as to the character of suspected roots. We may add, too, that any value of x which renders a polynomial negative, will render it positive if all the signs be changed. (See the note at the end, p. 247.) It may be useful to the learner to mention here that the positive roots of an equation may be changed into negative, and the negative into positive, by changing the sign of the second term of the equation, and also the signs of all the alternate terms after the second; taking care, however, to supply the places of the absent terms by zeros: thus the roots of the last equation are the same in value as the roots of x+2x3+0x2-3x-20—0, but opposite in sign. 96. When the first figure of one of the roots of an equation, has been found, the remaining figures may be obtained by Horner's process, which we shall now explain. (1) The first figure of a root of the equation a3+3x+2x71=0 is 3, (page 209), required the remaining figures. Hence the transformed equation (1) page 205, is x13+12x12+47x'—11=0...... (1), the first figure of the root of which is the second figure of the root of the proposed, as already explained. If this first figure were known, we should have to repeat with it the process above, using now the new coefficients, 1 12 47 -11, in conjunction with the new figure. The 47, taken as a trial divisor of the 11 (changing its sign), serves to suggest this new figure: the quotient is 2: hence, repeating the operation, 1 ཡ -11[2 9.888 -1.112 Hence the transformed equation (2), page 206, is a3+12-6x+51·92x"-1·112=0......(2), the first figure of the root of which is the third figure of the root to be developed, and 51.92 is the trial divisor of 1·112 for suggesting this figure; the quotient being 02, we have for the next step so that the next transformed equation is 3+1266x2+52·4252x068552-0......(3); the first figure of the root of which is the fourth figure of the root in the process of development; and for this fourth figure the trial divisor 52.4... of 0685... gives '001. And in this way may the operation be carried on to any extent of decimals; the root thus far determined is 3.221. The learner will perceive that what we have called the trial divisor in each step, is that which is converted into the true divisor by receiving a correction which becomes more and more insignificant at each succeeding step of the work: in the second step, the trial divisor was 47, and the true divisor 49.44; the difference, even in this second step, being too small to render any change in the suggested figure necessary; in the next step, the trial divisor was 51.92, and the true divisor 52-1724, the difference between the two being still smaller; in the step next following, which we have not completed, the trial divisor is 52-4252, which, on account of the continually diminishing values of the successive root-figures, we may be sure will differ still less from the true divisor; and it is easy to see that for the same reason the corrections of the trial divisors, becoming continually of less importance, may, after a certain stage, be wholly neglected, and the work thenceforward be made to assume the character of common division. Thus, in the operations above, it is evident that even after the third step, that is, after developing the root no further than 3.22, two or three more figures may be correctly obtained without taking any account of the subsequent corrections; for as the figure next suggested is so small as '001, it is easy to foresee that the first three figures of the trial divisor, 52-4252, which the third step supplies, cannot possibly be affected by these subsequent corrections. Taking, therefore, 0685 as a dividend, and 524 as a divisor, we may, for two or three more figures, proceed thus: 52.4] 0685 [001307 ⚫0524 161 157 4 4 We have here employed the contracted method of division explained in all books on arithmetic; and, as we shall presently see, all the figures in the quotient are correct; the root, as far as six decimals, being 3·221307. From these details the learner will perceive why the term trial divisors was employed, and the propriety of the designation. They not only suggest the root-figures in the earlier steps of the work, but, by approaching nearer and nearer to the true divisors, may be actually substituted for them at the close. In order, however, that the entire process may the better harmonise with these closing steps, the sign of the final term in the original equation should be changed, or the term itself transposed, and the number placed under it subtracted, instead of added, so that the operations in the last column of the work may be uniformly those of subtraction instead of addition. Adopting this plan, uniting the steps here exhibited in a detached form, and checking the accumulation of useless decimals in the several columns, after the method of contracted division, the work will take the following compact form, in which the Egyptian figures 1, 2, 3, &c. point out the coefficients of the successive transformed equations (1), (2), (3), &c. * At this stage of the process we shall check the further increase of decimals In the last column, limiting the number to the six places now reached; the preceding The transformations marked 1, 2, and 3, are the same as those exhibited in a detached form above, and marked (1), (2), and (3); the transformation 4 is the transformation 5 is the transformation 6 (5); x3+12·66x2+52·451x-016114=0......(4); and the transformation 7x+12.66x2+52·455x- root. 1 1 -29 -27 6 [5.372281 (3) Required a root of the equation 12x3-120x+326x— 127=0. columns must therefore be so curtailed as to exclude everything not tributary to the last column within this extent, and to include everything that is. * The learner will take notice that, in consequence of the contractions employed in the development of the root above, the coefficients of the quadratic here obtained are only close approximations to the true coefficients: it is easy to see, however, that no correction of the decimals can influence the conclusion here arrived at. |