Multiplying by 3x, to make the first term a square, the equation is 36x-360x+978x-381x=0 6x2+(360x3-978x2+381x), 6x-30x{x(-78x+381)}.. <5; (see p. 248); consequently the equation has no root so great as 5: a glance at the coefficients shows, however, that one root lies between 0 and 1; that is, that the first figure of a root is O in the units' place; hence, taking 326 as the trial divisor of 127, and observing that, on account of the -120, the true divisor will be less than this, the root-figure 4 is suggested, which is found to succeed, as 5 would introduce a change of sign in the final term; the development of the root is, therefore, as follows:127 [-46567742 111.968 12 -120 326 4.8 -46.08 1 13.768032 1 2 1.263968 The final transformed equation, of which the roots are all diminished by the root now determined, is 12x-103-38x2+222·06x=0; one of these roots being of course x=0, as the number by which all have been diminished, being itself a root, reduces that root to zero; hence we have the quadratic 12x2-103.38x+222·06=0; which is such that four times the product of the extreme terms differs from the square of the middle term only in the decimals; and as on account of the decimal contractions in the preceding process of development, the decimals in the coefficients of the quadratic are but approximations to the true decimals, it would not be safe to conclude whether the two remaining roots are real or imaginary: all we can infer is, that a very slight change in the coefficients of the proposed equation would cause two of its roots to become equal. The first two figures of each of the roots of the quadratic is 4·3, so that when increased by ·46..., the first two figures of the remaining roots of the proposed equation, if real, would appear to be 47..., we may be certain that the first figure of each is 4. If we were to enter upon the development of one of the roots, commencing with this first figure, carrying on the process as in the operation above, on the presumption that we are following up a real root, we should find after two or three steps that the final column of the work, instead of tending to terminate in zero, would tend towards a finite quantity which could not be further diminished: we should then infer that the roots presumed to be real are in fact imaginary. 97. It was shown above that no root of the equation can be so great as 5; that is, that all the real positive roots lie between O and 5. If we were to diminish the roots by 4, 3, 2, 1, in succession, we should not arrive at any change of sign to indicate the presence of a real root: the only root we could detect would be that between 0 and 1, just developed. A change of sign occurs only when a single root, or some odd number of roots, lies between the two numbers substituted. No change takes place when an even number of roots lies between the numbers employed; so that without some other means of putting a pair of suspected roots to the test, we cannot say whether they are real or imaginary: the plan just alluded to of entering on their development as if they were real, will always succeed in settling the doubt. But the learner who desires to be fully acquainted with the researches of algebraists on the theory and solution of numerical equations, may consult the works, already referred to, by the author of this Introduction, more especially the "Analysis and Solution of Cubic and Biquadratic Equations:" a subject which he will find to be as attractive and interesting as it is practically useful and instructive. To the present brief sketch, we shall only add that when one root of an equation is determined as above, if the original poly nomial be divided by x minus that root, the quotient equated to zero will be an equation of lower degree containing the remaining roots, and with this depressed equation we may proceed as with the original; but in the work just referred to, instructions are given for finding the places of all the real roots of an equation at the outset; after which the roots themselves may be developed, as above, one after another: and the use of the preliminary resolution of the polynomial into a pair of factors, as in the foregoing examples, in this search after the places of all the roots of an equation, may be seen in the tract on the "Analysis of Equations," forming Part I. of the "General Principles of Analysis." EXAMPLES FOR EXERCISE. It is required to determine a positive root of each of the following equations:— Find a negative root of each of the following equations:(9) x1-12x2+12x-3=0. (10) x3-3x+6=0. (11) x3-5x+3x+48=0. (12) x1+x2-8x-15=0. (13) x1—2x3+3x-20=0. (14) x1+x3+x2+x—1=0. (15) 2x1+3x3-6x-8=0. 98. We here terminate the chapter on the solution of equations; to which a brief space has been thus devoted, in the hope of exciting in the learner a disposition to inquire further into this important department of algebra, by familiarising him with the principle of Horner's method, and with the mode of decomposing a polynomial into factors. The illustrations we have given of the use and simplicity of such decompositions have * In developing a root of an incomplete equation, after the first figure of it is found, the absent terms must be replaced by zeros: thus the present equation, in a complete form, is x4+0x3+x2-8x-15=0. necessarily been very limited; but enough has been shown to enable the learner to perceive that, from the choice of forms at his disposal, factors to suit any case may always be selected: any term we please may be converted into a square by multiplying the polynomial by the suitable factor, and any term we please may afterwards be expunged from under the radical; since negative as well as positive powers of x may be introduced into the rational part. For instance, if example (2), page 209, had been 3+3x2+2x-73=0, or 3x3 +9x2+6x-219=0, we should have had 3x√(-3x-6x+219), which gives 5 for the superior limit of the positive roots; but the step 3#—219 v{—3z—6*+ (?l9)}, shows at once, without any nice calculation of the last term under the radical, that 4 is a superior limit; and not only so, but that a root is much nearer to 3 than to 4; in fact, if the expression under the radical be examined more minutely, we shall find that even 3.3 is a superior limit, and thus that the second figure is also suggested. Again, by decomposing in two distinct ways, both superior and inferior limits may often be readily found; for example, let the equation be x3+11x2-102x+181-0, the analysis of which has been hitherto attended with some difficulty: changing it into x+11x3-102x2+181x=0, we have, for one form of decomposition, the factors showing that a root cannot exist in the interval 0, another form of decomposition is : but 529 showing that no root can be so great as 4: it turns out that two roots of the equation lie between 3 and 4: they are 3.213... and 3.229..., as shown in Part I. of the "General Principles of Analysis," page 34. But we cannot dwell further upon these particulars, although, from the fertility of the principle which has led to them, there is room for a much more ample discussion of it, as a reference to the tract just quoted will show. (See also the note, p. 247.) 99. NOTE.-It may be interesting to the readers of other works on Algebra to be here shown how readily the formulæ of Ferrari and Descartes, for the general solution of a biquadratic equation of the form 4+pa2+qx+r=0, are obtained from the method of decomposition employed in the present chapter: thus x2±√{−px2-qx−r}, ... if I be determined so as to fulfil the condition (2k—p) (k2—r)=?2 (1), the expression under the radical will be a complete square, of which the root is √2k-p. x-√k2—r; so that the component quadratics, of the original equation, are and these are the quadratics furnished by the method of Ferrari. Again, a2+}p±√{-qx+\p2-r}, and to make the quantity under the radical a square, the condition is z3+2pz2+(p2-4r)z-q2=0....(2). The solution of this cubic makes known the value of k in the component quadratics q 2k x2+1( p+k2) — ( kx— %) =0, as in the solution of Descartes. 452-3.) The equations (1), (2), are both of them cubic equations; so that a biquadratic equation of the proposed form can be solved generally, provided we can solve generally a cubic equation: but this we have no means of doing in finite terms; that is, without infinite series, when certain numerical values are put for the symbolical coefficients. By imitating the decomposition above, a complete biquadratic equation may be solved by aid of a reducing cubic, as shown in the author's tract on the "Analysis of Numerical Equations," before referred to; but every complete equation may be reduced to another deprived of its second term by an easy transformation. It is much to be feared, however, that the discovery of formulæ for the solution of equations, with general symbols for the coefficients, will ever continue to baffle the efforts of analysts. Beyond equations of the second degree, but little advance has been hitherto made in this direction; and the prosecution of an inquiry found to be so barren of useful results-though the greatest minds have engaged in it, is now pretty generally abandoned by mathematicians, whose attention has of late been mainly directed to the more manageable subject of the solution of numerical equa (See the "General Theory of Equations," pages |