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divided by it, and the quotient is x2+2x+2=0, whose roots are -1±√-1; the five roots of the proposed equation are, therefore, 2, 3, -4, -1+ V-1, -1-√-1.

564. If the highest power of the unknown quantity has any coefficient prefixed to it, let the equation be assumed of the form (nx+a)P=0, and substitute 2, 1, 0, −1, −2, successively for x, as in the former instance.

Then, as before, the divisors of the several results, arising from this substitution, will be the terms of the arithmetical series

2n+a, n+a, a, -n+a, and -2n+a;

where the common difference n must be a divisor of the first term of the equation, or otherwise the operation would not succeed.

Hence, in this instance, the progressions must be so taken out of the divisors, that their terms shall differ from each other by some aliquot part of the coefficient of the first term.

Therefore, if the terms of these series, standing opposite to 0, be divided by the common difference, the quotient thus arising, taken in + and -, according as the progression is increasing or decreasing, will generally be the roots of the equation.

It is necessary to continue the series 2, 1, 0, -1, -2, far enough to show whether the corresponding progression may not break off, after a certain number of terms; which it never can do when it contains a real rational root.

Ex. 3. Given 2x3-3x2+16x-24-0, to find the roots of the equation, or values of x.

Substituting 2, 1, 0, -1, -2, successively, for x, as in the former case, we shall have

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Where the progression is ascending, the number to be tried is, therefore, , which is found to be a root of the equation.

Let the given equation be divided by x-3, and the quotient is 2x2-16-0, whose roots are +2√2; the three roots of the proposed equation are, therefore,,+2/2, −2√2.

Ex. 3. Given x4x3-29x2-9x+1800, to find the roots of the equation.

Ans. 3, 4, 3, and -5. Ex. 5. Given x2-4x3- 8x+32=0, to find the roots of the equation, or values of x.

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Ans. x 2, or 4; or 1-3. Ex. 6. Given x3-5x2+10x-3=0, to find the integral root of the equation.

Ans. 2. Ex. 7. Given x1--§x3 +x2+82x-60=0, to find the integral roots of the equation. Ans. 5, and −3. Ex. 8. Given x3-9x3+8x2-72=0, to find the roots of the equation, or values of x.

3

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

§V. RESOLUTION OF EQUATIONS BY NEWTON'S METHOD

OF APPROXIMATION.

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 irrational, recourse must be had to a different process, as they can then only be obtained 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,

x+pxm-1+9xm--2+rxm-3+.. sx2+tx+u=0. (1). Then, supposing a to be a near value of x, found by trial, and 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+2)+p(a+z)-1+..s(a+z)2+t(a+z)+n=0; which last expression, by involving its terms, and taking the result in an inverse order, may be transformed into the equation

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P+Qz+Rz2+Sz3+ ··· +z”=0 (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

P=a+pam+9am-2+

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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 coeffi cients of z, it will likewise appear, that

Q=mam-1+m(m−1)paTM--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 z2, z3, z1, &c. which are all, successively, less than z, be neglected in the transformed equation, we shall have

a+pa-1+

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+ta+u.

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P+Qz=0, or z ma ¿m--1+ (m—1) pam-2 +; .+t And, consequently, if the numeral value of this expression be calculated to one or two places of deci

mals, and put equal to b, the first approximate part of the root will be ż=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 be

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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 Resolution des Equations Numériques. See, for farther particulars relating to this, and other methods, BONNYCASTLE'S Alge» bra, or BRIDGE's Equations.

Ex. 1. Given x3+2x2-8x=24, to find the value of x by approximation.

Here by substituting 0, 1, 2, 3, 4, successively for r 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. x2=a3+3a2z+3a22 +23

Then

2x22a2+4az+2x2
-8x=-8a-8z

=90.

And by rejecting the terms z3+3az2 +2z2, (Art.569), as being small in comparison with z, we shall have a3+2a-8a+3x2z+4az-8x=24;

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

Again, if 3.09 be substituted for a, in the last equa

tion, we shall have z=

24-a3-2a2+8a

24-29.503629-19.0962+24.72

3a2 +4α-8

28.6443+12.36

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