III. When several decimal places are found in the root, the operation may be shortened according to the method of contractions indicated in the examples. 314. Let us now work one example in full. Let us take the equation of the third degree, x3-7x+7 = 0. By Sturm's rule, we have the functions (Art. 299), therefore, the equation has three real roots, two positive and one negative. To determine the initial figures of these roots, we have hence there are two roots between 1 and 2, and one between I ++++ no variation. 'Therefore the initial figures of the two positive roots are 1.3, 1.6. Iet us now find the decimal part of the first root. The operations in the example are performed as follows: 1st. We find the places and the initial figures of the posi tive roots, to include the first decimal place by Sturms' theorem. 2d. Then to find the decimal part of the first positive root, we arrange the co-efficients, and perform a succession of trans formations by Synthetical Division, which must begin with the initial figures already known. We first transform the given equation into another whose roots shall be less by 1. The co-efficients of this new equation are, 1, 3, 4 and 1, and are all, except the first, marked by a star. The root of this transformed equation, corresponding to the root sought of the given equation, is a decimal frac tion of which we know the first figure 3. We next transform the last equation into another whose roots are less by three-tenths, and the co-efficients of the new equation are each marked by two stars. The process here changes, and we find the next figure of the root by dividing the absolute term .097 by the penulti mate co-efficient 1.93, giving .05 for the next figure of the root. We again transform the equation into another whose roots shall be less by .05, and the co-efficients of the new equation are marked by three stars. We then divide the absolute term, .010375 by the penultimate co-efficient, - 1.5325, and obtain .006, the next figure of the root: and so on for other figures. In regard to the contractions, we may observe that, having decided on the number of decimal places to which the figures in the root are to be carried, we need not take notice of figures which fall to the right of that number in any of the dividends. In the example under consideration, we propose to carry the operations to the 9th decimal place of the root; hence, we may reject all the decimal places of the dividends after the 9th. The fourth dividend, marked by four stars, contains nine decimal places, and the next dividend is to contain no more. But the corresponding quotient figure 8, is the fourth figure from the decimal point; hence, at this stage of the operation, all the places of the divisor, after the 5th, may be omitted, since the 5th, multiplied by the 4th, will give the 9th order of decimals. Again: since each new figure of the root is removed one place to the right, one additional figure, in each subsequent divisor, may be omitted. The contractions, therefore, begin by striking off the 2 in the 4th divisor. In passing from the first column to the second, in the next operation, we multiply by .0008; but since the product is to be limited to five decimal places, we need take notice of but one decimal place in the first column; that is, in the first operation of contraction, we strike off, in the first column, the two figures 68: and, generally, for each figure omitted in the second column, we omit two in the first. It should be observed, that when places are omitted in either column, whatever would have been carried to the last figure retained, had no figures been omitted, is always to be added to that figure. Having found the figure 8 of the root, we need not annex it in the first column, nor need we annex any subsequent figures of the root, since they would all fall at the right, among the rejected figures. Hence, neither 8, nor any subsequent figures of the root, will change the available part of the first column. In the next operation, we divide .000141586 by 1.4772, omitting the figure 8 of the divisor: this gives the figure 9 of the root. We then strike off the figures 4.0, in the first column, and multiplying by .00009, we form the next divisor in the second column, - 1.4769, and the next dividend in the 3d column, .000008663. Striking off 5 in this divisor, we find the next figure of the root, which is 5. It is now evident that the products from the first column, will fall in the second, among the rejected figures at the right; we need, therefore, in future, take no notice of them. Omitting the right hand figure, the next divisor will be 1.476, and the next figure of the root 8. Then omitting 6 in the divisor, we obtain the quotient figure 8: omitting 7 we obtain 6, and omitting 4 we obtain 7, the last figare to be found. We have thus found the root x = 1.356895876....; and all similar examples are wrought after the same manner. |
