CHAPTER III

THE EQUATION OF THE FOURTH DEGREE

776. Equations of the fourth degree which can be solved by means of the quadratic equation have for the most part been solved in Chapter VII, Book IV. Their solution offered no difficulty.

If an equation of the fourth degree has one rational root, it may be found by the same method used in connection with the cubic equation, 771. If this root is r, then the equation of the fourth degree can be divided by x-r and depressed to an equation of the third degree, which we have already learned to solve in its most general form. If the equation of the fourth degree has two rational roots, then it can be depressed to an equation of the second degree, which can be readily solved.

=

EXAMPLE.-Solve the equation x1 + 4x3 6x2+ 24x 72 0. 3 and combinations of these

The factors of 72 are 2, by twos, threes, and fours.

2, 3,

On trial it is found that 2 is a root of

the given equation, and on dividing it by x-2 the depressed equation is the cubic

x3 + 6x2 + 6x + 36 = 0.

The factors of 36 are 2, 2, 3, 3 and combinations of these by twos and threes. On trial it is found that - 6 is a root of the cubic. Dividing the cubic by +6, the depressed equation is

x2+60 or x=1 - 6.

Hence the roots of the given equation are 2.

EXERCISE CXV

The following equations are of the character just described; they have at least one or two rational roots. These may be found according to the rule illustrated in 771; then the remaining roots are to be found by depressing the equation as illustrated in 1776.

21. 26.c1

108.x3+323.xr 241.x + 60 0.

22. 36x 72x3 31.x2+67x+ 30 = 0.

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777. Resolvent Cubic.. -In order to solve the equation of the fourth degree in general, one must first reduce it to the solution of a cubic equation. Then, by means of the roots of this equation, which is called the resolvent of the given equation, we have to determine the roots of the given equation.

(1)

778. In order to solve the equation of the fourth degree,

ax +4 bx+6cx2 + 4 dx+e=0

where t, P, Q are to be determined. Equation (3) developed and arranged with respect to x is

(4) a2x2+4abx3 + (4b2+4at+2ac—4 P2)x2+(4bc+8bt—4PQ)x

+c2+4t2+4ct — Q2=0.

On comparing like powers of x in equations (2) and (4) we have

Eliminating P and Q from (5), (6), and (7) we have the resolvent cubic

(8) 4 t3 (ae ·

-4bd + 3e2) t + ace + 2 bed ad2- b'e - c3 = 0.

The biquadratic equation (3) is equivalent to the two equations:

(9) ax2+2(b-P)x+c+2t-Q=0,

(10) ax2+2 (b + P)x + c + 2 t + Q=0.

If

X1 and x are the roots of (9) and x, and are the roots of (10), then we have

(11) x+x=-2 (b — P),

a

x3 + x = − 2 (b + P).

a

Since there are three values of P corresponding to the three values of t, equation (11) includes the following systems:

Whence follow the values of the root x1, x2, X3, X4.

Since the P's in (12) are square roots, they may have positive as well as negative signs; the choice of signs which must be made is determined by the circumstance that the relations

(13) = (X3+ X3) Xz¥, + (x3 + x) XX2 = −

a

4 d

a

between the roots and the coefficients of equation (1) are satisfied under all conditions.

The relations in (13) are due to the following theorem, which is true for any rational integral equation in x (8796):

THEOREM. In any rational integral equation in x, the coefficient of whose highest term is unity, the coefficient P1 of the second term with its sign changed is equal to the sum of the roots.

The coefficient P2 of the third term is equal to the sum of the products of the roots taken two by two.

The coefficient P3 of the third term with its sign changed is the sum of the products of the roots taken three at a time in all possible ways; and so on, the signs of the coefficients taken alternately negative and positive, and the number of the roots multiplied together in each term of the corresponding expression of the roots increasing by unity, till finally that expression is reached which consists of the product of the n roots.

If all of the roots x1, x2, x3, x, are real, which can happen only in case all the P's are real, and consequently if the three values of t are real, then the given equation can be separated in three ways into two quadratic factors with real coefficients. The imaginary roots must of necessity be conjugates and enter in pairs, because an imaginary root arises from the extraction of a square root, and the root may be positive or negative. If x and x, are a pair of con

X1

are real, and equation (7) Hence one decomposition

jugate roots, then x, x, and also x, x has also in this case real coefficients. (and one only) in two quadratic equations with real coefficients is possible where the roots of the given equation are all or in part imaginary.

If, therefore, the resolvent cubic has three real roots, there is one of them which gives a real P and, on account of (7), a real Q. If the resolvent cubic has but one real root, then this real root must also furnish a real P; for only in such a case can there be a decomposition in two real quadratic factors which have real coefficients. In case the biquadratic equation has b= 0, then the resolvent cubic is

(c -- t) (ae (c + 2t)2) = ad2 and P = Va(t — c).

EXERCISE CXVI

The following equations of the fourth degree (134) may in at least one way be separated rationally into quadratic factors. The resolvent cubic has therefore at least one rational root, which can be determined simply in the usual way; the expression for P can be at once calculated, and the calculation of the roots of the given equation follows without any great difficulty.