The other divisors, positive and negative, of 15 are,

+3, +5, +15, −3, −5, —15.

The quotients of -15 by these divisors are, respectively,

—5, —3, −1, +5, +3, +1.

Adding +13, the coefficient of the term in x, to each quotient, the sums are, +8, +10, +12, +18, +16, +14.

The integer quotients obtained from the division of these sums by the corresponding divisors are *, +2, *, −6, *, *.

Adding -12, the coefficient of the term in x2, to each of these results, the sums are 10, -18.

The integer quotients obtained from the division of these sums by the corresponding divisors of -15 are, −2, +6.

Adding +2, the coefficient of the term in r3, to each of these results, the sums are 0, +8.

The substance of the preceding detail is commonly exhibited in a tabular form, thus:

Since the only column in which the result B'+A=0 is found, is that of the divisor 5, it is concluded that +5 is the only integer root of the proposed equation.

The root +5 being known, the equation is divisible by the factor x—5; effecting the division the quotient is 2x2-2x+3; therefore 2x2-2x+3=0. Whence r+-5 (Art. 104 and 120).

The proposed equation is thus completely resolved, its three roots being x=5, x={+}√−5, x=§—§√−5.

This method does not indicate how often the roots which are determined by it enter into the equation; to discover this the most obvious process is to divide the equation by the factors corresponding to these roots, each being taken once, and afterwards to examine, either by immediate substitution or by the preceding method, whether these roots still belong to the equation; if some of them verify the equation, it is certain that these enter at least twice into the original equation; by a new division it is determined whether they enter into it a third time, and so on.

Another process by which the number of repetitions of the same root is determined is supplied by the properties of derived polynomials; for if an equation, X=0, involves the factor (x-a)", the derived polynomial, X', ought to involve the factor (x-a)"-1, the polynomial X", derived from X', the factor (x-a)"−2, &c.

So that the number of the equations, X=0, X'=0, X”=0, . . . which are verified by the value x=a, will indicate how many roots equal to a are contained in the equation X=0.

253. All the integer roots, equal and unequal, being found, they may be taken from the equation (Art. 231); then, if any commensurable roots remain, they must be fractional.

The determination of the fractional roots depends on this principle :—If the first term of an equation has 1 for coefficient, and the other terms have integer coefficients, this equation cannot have any commensurable roots which are not whole numbers.

То prove this, let xTM+PxTM-1+Qxm−2. ... +Tx+U=0, be an equation in which P, Q... T, U express whole numbers.

a

Substituting for r a fractional number, reduced to the lowest terms, the proposed equation becomes

am-2

+P bm-1+Qfm-2 +T+U=0.

Multiplying all the terms by b-1, and transposing,

am

̄=—PaTM~'—QaTM-2b. . . . —TabTM-2-Ubm-'.

Now a, b being prime to each other, the first member of this equality is an irreducible fraction, while the second is a whole number. The equality is therefore impossible, and by consequence no fractional number can satisfy the equation.

a. If, after removing denominators, the coefficient of the first term is not 1, the equation can be transformed into another fulfilling this condition; for which purpose it is sufficient to make the unknown quantity, x, equal to a new unknown quantity, y, divided by the coefficient of the first term of the equation in x. Then it is evident that all the commensurable roots of the equation in x must have a fixed relation to the commensurable roots of the equation in y; and as this can have integer roots only, these can be determined by the preceding method, and from them the corresponding roots of the equation in x.

The transformation of the equation in x into the equation in y is effected as follows:

Let the proposed equation be Ax”+Bx¬¬1+Cx”-2+, &c. in which A, B, C .. are whole numbers.

.....

The coefficient of y" in this equation becomes 1, and the other coefficients are whole numbers if z=A.

In some particular cases z may be chosen less than A, by which means a transformed equation with smaller coefficients is obtained.

As a second example, let it be required to apply these principles to the discovery of the commensurable roots of the equation 4x+-11x2+7x— 6=0?

The integer roots might be first sought; but as the coefficients are small numbers, it seems preferable at once to transform the proposed equation into another of which the first coefficient shall be 1, and the other coefficients whole numbers.

Following the general rule, it is necessary to make x=2

but in the pre

2

sent instance it is sufficient to make x=2; substituting for r and reducing, it is found that

or 4x16-11x+7x-6=0,

or y-11y2+14y-24=0,

the only commensurable roots of which equation are whole numbers.

Since the equation is not satisfied by the substitution for y −1, +1 and −1 are not roots of the equation.

Process for finding the roots.

of +1 and

The conclusion to which this calculation leads is that +3 and 4 are roots of the equation in y.

Dividing the equation in y by the product (y-3) (y+4) the quotient is y2−y+2=0;

whence y= √−7.

The four values of y being determined, the relation x=2 gives the corresponding values of x, viz.

x=3, x=-2, x=}+{√−7, x=÷¬{√−7.

When the equation whose integer roots are sought wants any of its terms, it is necessary to consider the coefficients of these terms as equal to zero; consequently the quotients to which these coefficients ought to be added may be immediately subjected to new divisions; this has been done in the case of the equation y*-11y2+14y-24=0, which wants the term in y3.

OF THE LIMITS OF THE ROOTS OF EQUATIONS.

254. The process which has been followed in determining the commensurable roots of an equation consists in the employment of a series of trials so ordered as certainly to discover all the roots of that kind which can exist in an equation.

It is also by trials that the approximate values of the incommensurable roots are obtained.

In investigating the rules which ought to direct these trials, the first problem is, to find the limits of the roots; in other words, to determine two numbers between which are comprised all the positive roots of the equation, and two numbers between which are comprised all the negative

roots.

a. To determine a superior limit of the positive roots, suppose, first, that the proposed equation X=0 is composed of certain positive roots, and that these are followed by certain negative roots, all of a less degree in x than the terms which are positive.

Let Y denote the assemblage of positive terms,
Y' - the assemblage of negative terms,
and the least power of x involved in Y;
Y Y'

then XY-Y'=x |

In the quotient of Y by a no term ought to contain z in the denominator, but in the quotient of Y' by 2 all the terms must contain x in the denominator; making, therefore, z to begin with any positive value and to

Y

increase, the quotient ought to increase, or, if Y is a monomial, to remain

Y'

constant, while necessarily diminishes.

It being, therefore, found that a positive value, z=1, renders positive the polynomial X or Y-Y', it is certain that by substituting in X greater values of x than 1, the results must be always positive, and must go on increasing to infinity; above there cannot, therefore, be any number which renders Y-Y'=0; and consequently I is a superior limit of the positive roots.

Whence it is to be concluded, that if in X are successively substituted for x increasing positive numbers until a positive result is obtained, the number which gives this result is the limit sought.

b. Suppose, next, that the equation contains positive mixed with negative terms, which is commonly the case; the first term of an equation always either is or can be rendered positive; combine with it the positive terms which occur before the first negative term, and (neglecting the other positive terms) compare this sum with that of the negative terms of the equation. Then the preceding reasoning proves that a superior limit of the positive roots will be obtained by determining a positive value of x, which renders the first sum greater than the second; and this is easily accomplished by substituting for x increasing numbers beginning from 0.

In the case of the positive and negative terms intermingled, it is often possible to separate the first member of the equation into several parts, in each of which care is to be taken to put one or more positive terms followed by negative terms of a lower degree; then determining as before, by successive trials, one positive value, departing from which all these parts are positive, this value may be taken for the limit required.

The terms of the equation may be separated into parts in different ways, and different limits may be obtained in consequence. Of these the lowest is always to be chosen.

c. In all cases the superior limit has this property, that if greater numbers are substituted in the equation the results are positive, and augment (if the numbers are increased) to infinity. The first condition is the only one necessary in order that I may be a superior limit of the positive roots; for it results from a theorem (for which see Article 254. a.) that every number greater than the greatest positive root gives a positive result.

As an example of the application of these principles, let it be required to find a superior limit of the roots of the equation

x+-3x3+2x2-3x-8=0?

Comparing the first term of this equation with the negative terms, the polynomial -3x3-3x-8 is to be considered.

Making a successively 0, 1, 2, 3, 4, the value of the polynomial is -8, -10, -22,-17, +44. Whence, as x=4 gives a positive result, 4 is a superior limit of the roots of the proposed equation.

But since the first member of this equation is the sum of the two quantities x-r3 and 2x2-3x-8, of which the first becomes 0, and the second +1, on the hypothesis of x=3, it appears that 3 is also a superior limit of the positive roots of the proposed equation.

2d Example. It is required to find a superior limit of the positive roots of the equation

x7+8x+2x5-10x+-40x3+10x2-14x-100=0?

The first member of this equation can be separated into the parts,
(x2-100), +(8x+2x5-10x+-40x3), +(10x2-14x),

or (7-100), +8x3 (x3+4x2-4x-5), +10x(x−7).

Making r successively 0, 1, 2, it is found that each part is positive, for x=2. Therefore 2 is a superior limit of the positive roots of this equation.

255. Newton employed a process by which a more approximate limit is frequently found than by other methods; it consists in finding a number such that if it is subtracted from the rootsof the proposed equation the result is a transformed equation whose terms are all positive. Then it is evident that no positive value can satisfy this transformed equation, and that by conse quence the number which is to be found necessarily exceeds the greatest positive root of the proposed equation. Let this unknown number be l; to diminish all the roots of equation X=0 by 1, it is necessary to make x=1+y. The result is a transformed equation in y, which may be put under the form

L+Ly+zL ̋y2+÷L""'y3 .... +y=0.

L, L',L",... representing the polynomial X and its derivatives X′, X” in which x is replaced by l.

Now it is required that all the coefficients of the equation in y shall be positive; consequently the question is reduced to this, to find a value of r which shall render positive all the quantities X, X', X", &c. ?

The quantity X being of the degree m, X' is of the degree m-1, X" of the degree m-2... ., and so on to the last quantity which does not contain a, and which is essentially positive.

It is convenient, therefore, first to determine a value of r such as shall render the last term but one positive, which can be easily done, since this term is of the first degree in x.

Then this value is augmented, if necessary, until it renders the preceding term (or term in ao) also positive; and this process is continued up to the term X.

For convenience it is usual to employ whole numbers only in these trials.

Let it be required to find by this method a superior limit of the positive roots of the equation

x3+x^—4x3—6x2-7000x+800=0?

X""" and X"" are positive, for r=1;

X" is positive, for x=2;

X' is positive, for x=7.

As the value x=7 renders X also positive, it is concluded that 7 is a superior limit of the positive roots of the proposed equation.

In these trials it is never necessary to return to the preceding derived polynomials. Suppose, for example, that, commencing with the polynomial of the first degree in x, and ascending step by step to X", a value x=l has been found which renders positive all the polynomials X", X"", X"""' . then, if x is augmented by being changed into x+h, as the polynomials derived from X" are X", X"""" it is evident that X" will become X"+X"h+X"""'h2+ .. Consequently, when the value x=l renders positive, it is certain that a value greater than I will not only

X", X""

render X" positive, but also augment its value.

256. To determine an inferior limit of the positive roots of an equation, make in the proposed equation, and represent by l the superior limit of the positive roots of the transformed equation. Then it is evident that

1

y

ī will be a quantity less than the least positive root of the proposed equation (Art. 233); consequently this quotient may be taken for the limit sought.