x2+x+3=0; whence x=

thus the equation is completely resolved.

3. For a third example, take the equation

2

x2+5x+6x-6x-15x3-3x2+8x+4=0;

the derived polynomial is

7x+30x+30x1 — 24x3-45x2-6x+8 ;

and the common divisor is

x1+3x3+x3-3x-2.

The equation +3x3+x3-3x-2=0 cannot be resolved directly, but by applying the method of equal roots to it, that is, by seeking for a common divisor between its first member and its derived poly. nomial, 4x3+9x2+2x-3, we find a common divisor, x+1; which proves that the square of x+1 is a factor of x2+3x3+x2-3x-2, and the cube of x+1, a factor of the first member of the proposed equation.

Dividing a1+3x+x2-3x-2 by (x+1)2 or x2+2x+1, we have x2+x-2, which placed equal to zero, gives the two roots x=1, -2, or the two factors x-1 and x+2. Hence we have

x2+3x3+x2-3x-2=(x+1)2(x−1) (x+2).

Therefore the first member of the proposed equation is equal to (x+1)(x-1)2(x+2)2;

or the proposed equation has three roots equal to-1, two equal to +1, and two equal to -2.

Take the examples,

1st.

2d.

27-7x+102+22x2-43x3-35x+48x+36=0,

(x-2)2(x-3)(x+1)=0.

x-3x+9x5-19x4+27x3-33x2+27x-9=0,

(x-1)(x2+3)=0,

286. When, in the application of the above method, we obtain

an equation D=0, of a degree superior to the second, since this equation may itself be subjected to the method, we are often able to decompose D into its factors, and in this way to find the different species of equal roots contained in the equation X=0, and the number of roots of each species. As to the simple roots of X=0, we begin by freeing this equation from the equal factors contained in it, and the resulting equation, X'=0, will make known the simple roots.

CHAPTER VII.

Resolution of Numerical Equations, involving one or more Unknown Quantities.

287. THE principles established in the preceding chapter, are applicable to all equations, whether their co-efficients are numerical or algebraic, and these principles should be regarded as the elements which have been employed in the resolution of equations of the higher degrees.

It has been said already, that analysts have hitherto been able to resolve only the general equations of the third and fourth degree. The formulas they have obtained for the values of the unknown quantities are so complicated and inconvenient, when they can be applied, (which is not always possible), that the problem of the resolution of algebraic equations, of any degree whatever, may be regarded as more curious than useful. Therefore, analysts have principally directed their researches to the resolution of numerical equations, that is, to those which arise from the algebraic translation of a problem in which the given quantities are particular numbers; and methods have been found, by means of which we can always determine the roots of a numerical equation of any given degree. It is proposed to develop these methods in this chapter.

To render the reasoning general, we will represent the proposed equation by

in which P, Q... denote particular numbers, real, positive, or negative.

First Principle.

288. When two numbers p and q, substituted in the place of x in a numerical equation, give two results, affected with contrary signs, the proposed equation contains a real root, comprehended between these two numbers

The first member will, in general, contain both positive and negative terms; denote the sum of the positive terms by A, and the sum of the negative terms by B, the equation will then take the form

A-B=0.

Suppose pq, and that p substituted for x gives a negative result, and q a positive result.

Since the first member becomes negative by the substitution of p, and positive by the substitution of q, it follows that we have in the first case AB, and in the second A>B. Now it results from the nature of the quantities A and B, that they both increase as x increases, since they contain only positive numbers, and positive and entire powers of r; therefore, by making a augment by insensible degrees, from p to q, the quantities A and B will also increase by insensible degrees. Now since A, by hypothesis, from being less than B, afterwards becomes greater than it, A must necessarily have a more rapid increment than B, which insensibly destroys the excess that B had over A, and finally produces an excess of A over B. From this, we conceive that in the passage from A<B to A>B,

there must be an intermediate value for which A becomes equal to B, and the value which produces this result is a root of the equátion, since it verifies A-B=0, or the proposed equation. Hence, the proposition is proved.

In the preceding demonstration, p and q have been supposed to be positive numbers; but the proposition is not less true, whatever may be the signs with which p and q are affected. For we will remark, in the first place, that the above reasoning applies equally to the case in which one of the numbers p and q, p for example, is 0; that is, it could be proved, in this case, that there was at least one real root between 0 and q.

Let both p and q be negative, and represent them by -p' and -q'.

If, in the equation

m-1

...

Tx+U=0,

we change x into ―y, which gives the transformation

(−y)TM+P(—y)m-1+Q(−y)−2+

T(―y)+U=0,

it is evident that substituting -p' and -q' in the proposed equation, amounts to the same thing as substituting p' and q' in the transformation, for the results of these substitutions are in both cases

and

m-2

(—p')"+P(—p')TM-1+Q(—p')2+ ... T(−p')+U,

(−g')TM+P(—q′)m-1+Q(−q′) m2 + ... T(−q')+U ;

Now, since p and q, or -p' and -q', substituted in the proposed equation, give results with contrary signs, it follows that the numbers p' and q', substituted in the transformation, also give results with contrary signs; therefore, by the first part of the proposition, there is at least one real root of the transformation contained between p' and q'; and in consequence of the relation x=— -y, there is at least one value of a comprehended between -p' and -q', or p and q. This demonstration applies to cases in which p=0 or q=0.

Lastly, suppose p positive and q negative or equal to -q' by making 0 in the equation, the first member will reduce to its

last term, which is necessarily affected with a sign contrary to that of p, or that of -q'; whence we may conclude that there is a root comprehended between 0 and p, or between 0 and -q', and consequently between p and —q.

Second Principle.

289. When two numbers, substituted in place of x, in an equation, give results affected with contrary signs, we may conclude that there is at least one real root comprehended between them, but we are not certain that there are no more, and there may be any odd number of roots comprised between them. We therefore enunciate the second principle thus.

When an uneven number (2n+1) of the real roots of an equation, are comprehended between two numbers, the results obtained by substituting these numbers for x, are affected with contrary signs, and if they comprehend an even number 2n, the results obtained by their substitution are necessarily affected with the same sign.

4

To make this proposition as clear as possible, denote those roots of the proposed equation, X=0, which are supposed to be comprehended between p and q, by a, b, c, . . ., and by Y, the product of the factors of the first degree, with reference to x, corresponding both to those real roots which are not comprised between them and to the imaginary roots; the signs of p and q being arbitrary. The first member, X, can be put under the form

(x-a) (x—b) (x−c)... ×Y.

Now substitute in X, or the preceding product, p and q in place of r; we shall obtain the two results

Y' and Y" representing what Y becomes, when we replace a by p and g; these two quantities are necessarily affected with the same sign, for if they were not, by the first principle Y=0 would give at