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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

23+16.5z2+92.75z=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

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(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

Whence

Ar3+Br2+Cr=V, nearly.

V
́C+Br+Ar2°

(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+3Ar2y+3Ary2+Ay3

+ Br2+2Bry+By2 =V;
+Cr +Cy

or, arranging according to the powers of y, we have

Ay3+(B+3 Ar) 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

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where is the first figure of the root of equation (3), or the second figure of the root of equation (1).

Putting z 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'z=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 V very nearly equal to 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, .82793.74.008.

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

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If we multiply expression (6) by r, and add the product to the third coefficient, we shall have

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If we multiply expression (7) by r, and subtract the product from V, we shall have

V–Cr–Br2 — Ar3,

which is the quantity represented by V' in equation (3).

If we multiply the first coefficient A by r, and add the prod uct to expression (6), we shall have

B+2Ar.

(8.)

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 transformed 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 A 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.

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

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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

x3-3x2+5x=178

may be exhibited as follows:

C +5

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11

77 1st divisor.

3.625-2d dividend.

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Having found one root, we may depress the equation

x3-3x2+5x-178-0

to a quadratic by dividing it by x-4.5388. We thus obtain
22+7.5388x+39.2173=0,

where x is evidently imaginary, because q is negative and
greater than 22. See Art. 280.


After thus obtaining the root to five or six decimal places,
several more figures will be correctly obtained by simply di-
viding the last dividend by the last divisor.

Ex. 2. Find all the roots of the equation

x3+11x2-102x=-181.

The first figure of one of the roots we readily find to be 3.
We then proceed, according to the Rule, to obtain the root to.
four decimal places, after which two more will be obtained
correctly by division.

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20.636.343893 = 4th divisor.

The two remaining roots may be found in the same way, or
by depressing the original equation to a quadratic. Those

roots are,

3.22952
-17.44265.

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