CHAPTER II

THE CUBIC EQUATION

761. Definition of the General Equation of the nth Degree.We represent the rational integral expression

of the nth degree in x by the symbol f(x); then a rational integral equation of the nth degree in x is defined by the equation

In the following discussion the coefficients P1, P... are supposed rational unless otherwise stated.

NOTE 1.-If x" in the equation f(x)=0 is multiplied by a constant, po, the equation can be reduced to the form in (a) by dividing both members of the given equation by Po

NOTE 2.-If any of the terms of equation (a) are wanting, the equation is called incomplete, but if the terms are all present it is called complete.

762. A Root.-Any number, r, which substituted for x makes f(x) identically 0, i. e, ƒ(r)=0, is called a root of the equation f(x)=0.

763. The Existence of a Root.-It is assumed in this treatise that every rational integral equation of the nth degree, f(x) = 0, has one root, real or imaginary. The proof of this theorem is too difficult for a text of the scope of this work, and is given in treatises on the theory of equations.

=

764. If is a root of f(x) 0, then f(x) is exactly divisible For, if is a root of the equation f(x) = 0, then, by

by x r'.

2762, if is substituted for x, f(r)=0 and the theorem follows at once from the factor theorem, 101.

765. Conversely, if the first member of the equation f(x) = 0

is divisible by x — r, r is a root of f(x) = 0.

For, if is the quotient of f(x) by xr, we have

J(x) = (x − r) Q.

Substitute in this equation x = r and we have

f(r) = (r− r) Qxr = 0.

Therefore is a root of the equation f(c)

=0 since Qr == ∞.

766. Definition.-The cubic equation is a rational integral equation of the third degree and is written

767. The Simplest Case, Cube Roots of Unity. Solve the equation

the last two of which are imaginary numbers.

The last two of these, x, and x,, are the roots of the equation

x2 + x + 1,

therefore, their product x, x, is equal to unity; then we have

768. Since each of the imaginary cube roots of unity is the square of the other root, it is customary to represent the three cube roots of unity by 1, w, w2.

771. The Cubic Equation with One Rational Root.

Definition. A real root which is either an integer or a fraction is called a rational root or a commensurable root.

It was shown in 1764 that if r is a root of f(x)=0, then f(x) is divisible by x—r. Hence must be a factor of the constant term in f(x), (3101, note). Therefore to solve a cubic with one rational root, find the factors of the constant term and determine which of these substituted for x will make the first member of the equation 0; then divide the first member of the given cubic by a minus this factor (2764). The roots of the two factors equated to zero will be the roots of the equation.

EXAMPLE.-Solve the cubic a3 + 5 x2 + 7 x + 2 = 0, which has one rational root.

- 2

The factors of 2 are 1, 2, - 1,2; of these four numbers, only will make x3 + 5 x2 + 7 x+2=0. Therefore the first member of the given equation is divisible by x-(-2) or x+2 (8101, Ex. 1). Hence we have

x3 + 5 x2 + 7 x + 2 = (x + 2) (x2 + 3 x + 1) = 0;

therefore the roots of the given cubic are those of the factors equated to zero, or of

EXERCISE CXII

Find all the roots of the following cubic equations, which have at least one rational root:

In case the cubic equation does not belong to any of the classes previously discussed and does not have a rational root, then other considerations are necessary for its solution. In this case we begin

with what is called the reduced form of the cubic equation

where P and are arbitrary real numbers which may be integral or rational, positive or negative.

A. CARDAN'S SOLUTION

772. Reduction of the Cubic to the Reduced Form. The cubic written in the general form is

(b)

x3 + P1 x2 + P2 x + P3 = 0.

[8766]

The cubic (b) can be reduced to the form (c) by the transformation

Adding together the several terms on each side of the sign of equality, we find