Ex. 6. Given x3-5x2+10x-8=0, to find the integral root of the equation.

Ans. 2. Ex. 7. Given 1-8x3+x2+82x-60=0, to find the integral roots of the equation. Ans. 5, and 3.

Ex. 8. Given 3 − 9x3+8x2 - 72=0, to find the roots of the equation, or values of x.

Ans. x=-3, or -2, or 3; or 1-3.

SV. RESOLUTION OF EQUATIONS BY NEWTON'S METHOD OF AP

PROXIMATION.

565. The methods laid down in the preceding section will be found sufficient for determining the integral or rational roots of equations of all orders; but when the roots are irra. tional, recourse must be had to a different process, as they can then be obtained only by approximation; that is to say, by methods which are continually bringing us nearer to the true value, till at last the error being very small, it may be neglected.

566. Different methods of this kind have been proposed, the simplest and most useful of which, as LAGRANGE justly remarks, is that of NEWTON, first published in WALLIS'S Algebra, and afterwards at the beginning of his Fluxions—or rather the improved form of it, given by RAPHSON, in his work, entitled Analysis Equationem Universalis.

567. In order to investigate the above-mentioned method, let there be taken the following general equation,

xm+pxm¬1+qxm−2+rxm-s+. . sx2+tx+u=0. (1). Then, supposing a to be a near value of x, found by trial, and z to be the remaining part of the root, we shall have x=a+ z; and, consequently, by substituting this value for x in the given equation, there will arise

(a+z)m +p(a+2)m−1+ ..s{a+z)2+1(a+z)+u=0 ; which last expression, by involving its terms, and taking the result in an inverse order, may be transformed into the equation

P+Qz+Rz2+Sz3+...+20.. (2),

where P, Q, R, &c. are polynomials, composed of certain functions of the known quantities, a, m, p, q, r, &c. which are derived from each other, according to a regular law.

568. Thus, by actually performing the operations above indicated, or by referring to (Art. 539), it will be found that

Pam+pam-1+9am-2+... sa2+ta+u;

which value is obtained by barely substituting a for x in the equation first proposed.

And, by collecting the several terms of the coefficients ofz, it will likewise appear, that

Q=mam--i+m(m—1)pam-2+...+2sa+t;

which last value is found by multiplying each of the terms of the former by the index of a in that term, and diminishing the same index by unity.

569. Hence, since z in equation (2) is, by hypothesis, a proper fraction, if the terms that involve its several powers 22, 23, 24, &c. which are all, successively, less than z, be neglected in the transformed equation, we shall have

P+Q=0, or z=

am+pam1+

+ta+u mam-+(m—1)pam-2+.. +i

And, consequently, if the numeral value of this expression be calculated to one or two places of decimals, and put equal to b, the first approximate part of the root will be z=b, or x=a+b=a'.

Whence also, if this value of x, which is nearer its true value than the assumed number a, be substituted in the place of a in the above formula, it will become

which expression being now calculated to three or four places of decimals, and put equal to c, we shall have, for a second approximation towards the unknown part of the root

z=c, or x=a+c=a".

And, by proceeding in this manner, the approximation may be carried on to any assigned degree of exactness; observing to take the assumed root a in defect or excess, according as it approaches nearest to the root sought, and adding or subtracting the corrections b, c, &c. as the case may require.

570. A negative root of any equation may also be found in the same manner, by first changing the signs of all the alternate terms, (Art. 541), and then taking the positive root of this equation, when determined as above, for the negative root of the proposed equation.

571. In the practical application of this rule we must endeavour to find two whole numbers, between which some one root of the given equation lies; and by substituting each of them for x in the given equation, and then observing which of them gives a result most nearly equal to 0, we shall ascertain the whole number to which a most nearly approaches; we must then assume a equal to one of the whole numbers thus

found, or to some decimal number which lies between them, according to the circumstances of the case.

572. Since any quantity, which from positive becomes negative, passes through 0 (Art. 496), if any two whole numbers, n and n'; one of which, when substituted for x in the proposed equation, gives a positive, and the other a negative result ; one root of the equation will, therefore, lie between n and n'. This, of course, goes upon the supposition that the equation contains at least one real root.

573. It is necessary to observe, that, when a is a much nearer approximation to one root of the given equation than to any other, then the foregoing method of approximation can only be applied with any degree of accuracy. To this we also farther add, that, when some of the roots are nearly equal, or differ from each other by less than unity, they may be passed over without being perceived, and by that means render the process illusory; which circumstance has been particularly noticed by LAGRANGE, who has given a new and improved method of approximation, in his Traité de la Reselution des Equations Numériques. See, for farther particulars relating to this, and other methods, BONNYCASTLE's Algebra, or BRIDGE's Equations.

Ex. 1. Given 3+2x2-8x=24, to find the value of x by approximation.

Here by substituting 0, 1, 2, 3, 4, successively for x in the given equation, we find that one root of the equation lies between 3 and 4, and is evidently very nearly equal to 3. Therefore let a=3, and x=a+z.

And by rejecting the terms z3+3az2+2z2, (Art. 569), as being small in comparison with z, we shall have

a3+2a-8a+3a2z+4az-8z=24;

and consequently x=a+2=3.09, nearly.

Again, if 3.09 be substituted for a, in the last equation, we shall have z=

24-a3-2a2+8a

3a2+4a-8

24-29.503629-19.0962+24.72

28.6443+12.36-8

=.00364; and consequently x=a+z=3.09+.00364-3. 09364, for a second approximation.

And, if the first four figures, 3.093, of this number, be sub

stituted for a in the same equation an approximate value of will be obtained to six or seven places of decimals. And by proceeding in the same manner the root may be found still more correctly.

Ex. 2. Given 3x+4x3-5x=140, to find the value of x by approximation. Ans. x 2.07264. Ex. 3. Given (x2-9x3+8x2--3x+4=0, to find the value of x by approximation. Ans. x=1.114789. Ex. 4. Given x3+23.3x2-39x-93.3=0, to find the values of x by approximation.

Ans. x 2.782; or -1.36; or -24.72; very nearly. Ex. 5. Find an approximate value of one root of the equation x3+x2+x=90. Ans. x 4.10283. Ex. 6. Given 3-6.75x2+4.5x-10.25-0, to find the values of x by approximation.

Ans. x.90018; or - 2.023; or -5.627 ; very nearly.

END OF THE TREATISE ON ALGEBRA.

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