Hence, one root of V lies between .2 and .3, and one between .7 and .8. Consequently the initial figures of the two positive roots in the original equation, are 2.2 and 2.7.

NOTE. If we had found the initial figures of the two positive roots of V to be the same, we should have proceeded to transform V, and make similar trials with the result.

We are now prepared to find the roots of an equation to any degree of accuracy, by

HORNER'S METHOD OF APPROXIMATION.

463. In the year 1819, W. G. Horner, Esq., an English mathematician, published a most elegant and concise method of approximating to the roots of a numerical equation of any degree. The process consists in a series of transformations, the roots of each successive equation being less than the roots of the preceding equation by the initial figures of the preceding roots. But in making the several transformations, the initial figures are obtained by trial division, as in square and cube root, and not by substitutions, as in the last article.

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461. Let us take an equation in the general form, thus: X= =xm + Аxm-1 + Bxm¬2 + + Tx+U=0. (1). Let r represent the initial figure or figures of one of the real roots of this equation, as found by Sturm's Theorem, or otherwise. Now let the equation be transformed into another whose roots shall be less by r. Put x = r + y; we shall have

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V = ym + A'ym−1 + B'ym-2 + .... + T'y + U' = 0.. (2). In this equation y is supposed to represent a decimal, since r includes at least the entire part of the required root. Hence, the terms containing the higher powers of y are comparatively small; neglecting these, we have, approximately,

Denote the first figure of this quotient by s ; put y=s+z. Transforming (2) into another whose roots shall be less by s, we have V' = z1 + A'zm¬1 + B'zm−2+ .. + T'z +U" = 0.

where t is another figure of the required root. This process may be continued at pleasure, and we shall have, finally,

x = r + s + t + etc.

Hence, to solve a numerical equation of any degree, we first find by Sturm's Theorem, or otherwise, the number of real roots, and also the first figure or figures of each. We may then approximate to the value of any root by the following

RULE.-I. Transform the given equation into another whose roots shall be less by the initial figure or figures of the required

root.

II. Divide the absolute term of the transformed equation by the penultimate coefficient, as a trial divisor, and write the first figure of the quotient as the next figure of the root sought.

III. Transform the last equation into another whose roots shall be less than those of the previous equation by the figure last found; and thus continue till the root be obtained to the required degree of accuracy.

NOTES-1. The successive transformations required in obtaining any root may all be made in a single operation; and for the sake of perspicuity, the coefficients obtained in each transformation may be marked or numbered.

2. If a trial figure of the root, obtained by any division, reduces the absolute term X, and the penultimate coefficient X1, to the same sign, this figure is not the true one, and must be changed.

3. To obtain the negative roots, it will be most convenient to change the signs of the alternate terms of the given equation, and find the positive roots of the result; these, with their signs changed, will be the negative roots required.

4. If the penultimate coefficient, T", should reduce to zero in the operation, the next figure of the root may be obtained by dividing the absolute term, U', by the coefficient which precedes T', and extracting the square root of the quotient. For, if T' vanishes, we have, in the transformed equation,

EXAMPLES.

1. Given a 2x2 - 20x400, to find the approximate value of x.

By Sturm's Theorem we find that this equation has only one real root, the initial figure being 6. We now obtain the decimal part, to 2 places, as follows:

EXPLANATION.-We first transform the given equation into another whose roots are less by 6, using the method of Synthetic Division, explained in 443. The coefficients of the transformed equation are 16, 64, and 16, marked (1) in the operation. Dividing the absolute term-16, taken with the contrary sign, by the penultimate coefficient 64, we obtain .2, the next figure of the root.

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We next transform the equation whose coefficients are marked (1), into another whose roots are less by .2, the resulting coefficients being marked (2). Dividing 2.552 by 70.52, we obtain .03, the next figure of the root. The operation may thus be continued till the root is obtained to any required degree of accuracy.

2. Given x4x3-30x2-20x-20: 0, to find one value of x. By Sturm's Theorem, we find the initial figures of the two

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real roots to be 5 and 5. Changing the signs of the alternate terms of the equation, we obtain the decimal part of the negative root, by the following

EXPLANATION. We proceed as in the preceding example till we obtain the terms marked (2), in the operation. Dividing 10.2929 by 321.302, we obtain .03 for the next figure of the root.

At this point we commence to apply decimal contractions, according to the principles employed in the contracted method of cube root (243). Let it be observed, that each contracted term in the operation contains one redundant figure at the right.

Commencing with column IV, we have 21.8 x .03.65, which added to column III gives 148.49. Then 148.49 x .03 = 4.455, which added to column II gives 325.757. Then 325.757 ×.03 = 9.7727, which added to column I gives .5202. Again adding .65 to column III, we have 149.14. Then 149.14 x .03 = 4.474, which added to column II gives 330.23, after dropping one place. Again, adding .65 to column III gives 150 after dropping two places. In like manner we continue till the work is finished.

NOTE.-Observe, as a general rule, to contract the several columns, for each root figure, as follows: Column I, 0 place; column II, 1 place; column III, 2 places; column IV, 3 places; and so on.

NOTE.-Full solutions of the examples above may be found in the Key.