Of Commensurable Roots, and the equal Roots of Numerical

Equations.

197. HAVING made known the most important properties of algebraic equations, and explained the method of eliminating the unknown quantities, when several occur, I shall proceed to the numerical resolution of equations with only one unknown quantity, that is, to the finding of their roots, when their coefficients are expressed by numbers.*

I shall begin by showing, that when the proposed equation has only whole numbers for its coefficients, and that of its first term is unity, its real roots cannot be expressed by fractions, and consequently can be only whole numbers, or numbers, that are incommensurable. In order to prove this, let there be the equation

x2 + P x2¬1 + Qx12..... + Tx + U = 0,

in which we substitute for a an irreducible fraction; the equa

The first member of this last equation consists of two entire parts, one of which is divisible by b, and the other is not (98),

a

since it is supposed, that the fraction is reduced to its most simple form, or that a and b have no common divisor; one of these parts cannot therefore destroy the other.

198. After what has been said, we shall perceive the utility of making the fractions of an equation to disappear, or of rendering its coefficients entire numbers, in such a manner, however,

* There is no general solution for degrees higher than the fourth; properly speaking, it is only that for the second degree, which can be regarded as complete. The expressions for the roots of equations of the third and fourth degree are very complicated, subject to exceptions, and less convenient in practice than those, which I am about to give; I shall resume the subject in the Supplement.

that the first term may have only unity for its coefficient. This is done by making the unknown quantity proposed, equal to a new unknown quantity divided by the product of all the denominators of the equation, then reducing all the terms to the same denominator, by the method given in art. 52.

Let there be, for example, the equation

as the divisor of the first term contains all the factors found in the other divisors, we may multiply by this divisor and thus reduce each term to its most simple expression; we find then

y3 + anрy2 + b m2 np2 y + cm3 n3 p2 = 0.

If the denominators, m, n, p, have common divisors, it is only necessary to divide y by the least number, which can be divided at the same time by all the denominators. As these methods of simplifying expressions will be readily perceived, I shall not stop to explain them; I shall observe only, that if all the denominators were equal to m, it would be sufficient to make x =

The proposed equation, which would be in this case,

y

m

y3 + ay2 + b my + m2 c = 0.

It is evident, that the above operation amounts to multiplying all the roots of the proposed equations by the number m, since

199. Now since, if a be the root of the equation

we have

x2 + P xn−1 + Qx2-2 + Tx+U= 0,

....

it follows, that a is necessarily one of the divisors of the entire number U, and consequently, when this number has but few divisors, we have only to substitute them successively in the place of x, in the proposed equation, in order to determine, whether or not this equation has any root among whole numbers.

If we have, for example, the equation

as the numbers

X3 6 x2+27 x 38 = 0,

1, 2, 19, 38,

are the only divisors of the number 38, we make trial of these, both in their positive and negative state; and we find, that the whole number + 2 only satisfies the proposed equation, or that 2. We then divide the proposed equation by x2; putting the quotient equal to zero, we form the equation

=

the roots of which are imaginary; and resolving this, we find that the proposed equation has three roots,

x = 2, x = 2 + √·

15, x=2

15.

200. The method just explained, for finding the entire number, which satisfies an equation, becomes impracticable, when the last term of this equation has a great number of divisors; but the equation,

U=- an Pan-1

furnishes new conditions, by means of which the operation may be very much abridged. In order to make the process more plain, I shall take, as an example, the equation

x2 + P x 3 + Q x2 + R x + S = 0. The root being constantly represented by a, we have

a1 + Pa3 + Qa2 + Ra + S = 0,

Lastly, bringing P into the first member, making+P=P',

a

Putting together the above mentioned conditions, we shall perceive that the number a will be the root of the proposed equation, if it satisfy the equations

in such a manner, as to make R', Q', and P' whole numbers. Hence it follows, that in order to determine, whether one of' the divisors a of the last term S can be a root of the proposed equation, we must,

1st. Divide the last term by the divisor a, and add to the quotient the coefficient of the term involving x ;

2d. Divide this sum by the divisor a, and add to the quotient the coefficient of the term involving x2;

3d. Divide this sum by the divisor a, and add to the quotient the coefficient of the term involving x3;

4th. Divide this sum by the divisor a, and add to the quotient unity, or the coefficient of the term involving x1; the result will become equal to zero, if a is, in fact, the root.

The rules given above are applicable, whatever be the degree. of the equation; it must be observed, however, that the result will not become equal to zero, until we arrive at the first term of the proposed equation.*

201. In applying these rules to a numerical example, we may conduct the operation in such a manner as to introduce the several trials with all the divisors of the last term, at the same time. For the equation

All the divisors of the last term 15 are arranged, in the order of magnitude, both with the sign + and —, and placed in the same line; this is the line occupied by the divisors a.

The second line contains the quotients arising from the number 15, divided successively by all its divisors; this is the line S for the quantities

a

The third line is formed by adding to the numbers found in the

* It would not be difficult to prove by means of the formula for the

quotients given in art. 180, that the quantities.

S R' Q'

a

α α

taken with

the contrary sign, and with the order inverted, are the coefficients of the quotient arising from the polynomial

x2 + Px3 + Q x2 + R x + s

divided by xa, and which is, consequently,