Equations

Algebraic Solution of Equations.— The algebraic solution of the quadratic equation axt-i-bx-Fe=0 is well known. (See ALGEBRA). The algebraic solution of the cubic, due to Scipio Ferro and Tartaglia, and first published by H. Cardan in 1545, is known as aCardan's solution.* To effect it, first transform the gen eral cubic equation so that the second term shall be wanting. This done, we have x'+ax+ b=0. Putting x—y-Fr we obtain ?-1-3yr (y+$) +a+ a(y+z)-1-b=0, ye-F (3yz+a) (y+z) We may subject y and z to any second condi tion which is not inconsistent with x=y-i-e. It will be convenient to assume 3yz+a-0. Then or, substituting for z its as value —a/3y, we obtain y°4- by'=— 27 and Al ' --b=_-- b + 2 n* Since x=y+z, we have bi b x "=" — — • 2 4 27 2 N4 27 Since y' and Jr' have each three cube roots, it might seem as if y+z or x had altogether nine values. As the cubic has only three roots, this cannot be. Of the nine values, six are ex cluded by the relation 3yz-Fa3, which y and z must satisfy. Eliminating z between and 3yz+a'0, we get x=y— 3 where y has the three values obtained from the expression for y' given above. This last expression for z does not involve the difficulties of the first ex pression. If the numerical values of the coef ficients a and b are given, the numerical values of the roots may be obtained by substituting the values of a and b in the above expression for x. In any case, this mode of computing x is more laborious than Homer's method of approximation (explained below), but when all three roots of the cubic are real and dis tinct, an unexpected difficulty is encountered.

bs • 4 In this case - represents a negative num ber. As the square root of a negative number is a complex (imaginary) number, we are re quired to find the cube root of a complex num ber. But there exists no convenient arithmet ical process for doing this. Nor is there any way of avoiding the complex radicals and of expressing the values of the real roots by real radicals. This is the famous *irreducible case* in the solution of the cubic. Its interest is purely theoretical. The practical computor experiences no difficulty, for he can always find the values of x by the methods of approxima tion.

Since Cardan's time a great many different algebraic solutions of the cubic and also of the quartic have been given. They are brought together for convenient reference in L. Mat thiessen's (Grundziige der Antiken and Mo dernen Algebra,' Leipzig 1878. We proceed to give Euler's algebraic solution of the general quartic. By transforming it, bring it to the form x'-Faxt-i-bx-Fc=0. Assume the general expression for a root to be Squaring, Squaring again and simplifying, -Fv±iv)-8x V u V v V iv + 4 (uv+uw+trw) 0. Equating coefficients of this and the given quartic we have a=-2(n+v+w), b=-8V w.

c= (uv-Futo-l-vw): But —(u+v+w), (U2/±14W+ VW), --UM are the coefficients of a cubic whose roots are u, v, w. This cubic, called aEuler's cubic," is a a'-4c ' 2 16 Solving it, we have the values of u, v and w, and, therefore, the values of x. Of the eight apparent values of .r, four are excluded by the relation b==-8 \ruNTW'Fir. To solve the quartic by the present method we must, therefore, first solve cEuler's cubic," called the resolvent. When this resolvent has a rational root, then its other two roots can be expressed in terms of square roots and the quartic can be solved algebraically without the extraction of cube roots. All methods of solving algebraically the

general quartic depend upon the solution of some resolvent cubic.

Binomial equations of the form xn — 1= 0, or more generally, of the form xn—a=0, are known as cyclotomic equations, and can always be solved algebraically. They possess also many interesting properties. We . shall give a trigonometric solution and mention a few of theseproperties. Let xn=---a=r[cos (2kr +0) +i sin +0)1, where a may be a complex quantity, where k may be any integer, and where r and 0 are known from the value of a. (See TRIGONOMETRY). By De Moivre's heorem we obtain n_i 2k +0 . 2lor +0i COS - t.

yr By assigning to k any n consecutive integral values we obtain n distinct values for x and no more than n, since the n values recur in periods. These values are the roots required.

Among the properties of xn-13 are the following: It has no multiple roots; if r is a root, then any positive integral power of r is a root; if m and n are relatively prime, then stn-13 and xn-13 have no roots in common, except 1; if h is the highest common factor of in and n, then the roots of xh.-13 are common to .rni-13 and xn-1=0; if r is a complex root of x n being a prime number, then 1, r, • . . are the roots; the roots of xin-13 and .rn-1) satisfy the equation smn-1=0; xn —1=0 has always primitive roots, i.e., roots which are not also roots of unity of a lower degree than n. For the proofs consult Burnside and Panton, (Theory of Equations,' Vol. I. The theory of roots of unity is closely allied with the problem of inscribing regular polygons in a circle, or the theory of the "division of the circle." Consult P. Bachmann, 'Kreistheilung,' Leipzig 1872.

Solution by Approximation.— Of the vari ous methods which have been given for the solution of numerical equations, the most satis factory, all things considered, is the one known as "Horner's method." It is commonly used for finding incommensurable roots (i.e., such as in volve an interminable decimal which is not a repeating decimal), but it may be used also for finding commensurable roots (i.e., such as are in tegers or rational fractions). It is desirable here to begin with the theorem that a rational frac tion cannot be a root of an equation of the nth degree with integral coefficients, the coefficient of being unity. To prove this, let, if possible, k be a root of f(x) =0, where h and k are in tegers and — a fraction reduced to its lowest terms, and where cm=1. Substitute — for x, then multiply both members of the equation by k*-1, and we obtain, after transposing, k —ik — . . . — ankn This equation is impossible, since a fraction in its lowest terms cannot equal an integer. Hence cannot be a root. This being the case, it fol lows that all commensurable roots are exact divisors of an, for an is numerically the product of all the roots. We know that if (fx) is di-. visible by x—r, without a- remainder, r is a root. Hence we are enabled to find all com mensurable roots of numerical equations of the type now under consideration by testing in suc cession each factor of On. For instance, in the equation x' 8x' + 13x + 2-0 the factors of are ± I and ± 2. Taking the factor —2, we find that f(x) is exactly divisible by x + 2. The test for each of the three other factors yields a remainder. Hence —2 is the only com mensurable root.