all the positive coefficients which precede it, the greatest of the fractions thus formed, increased by unity, is a superior limit of the positive roots.

PmP(x-1)(x-1+x2+...+x+1)+Pm,

if we transform every positive term by this formula, and leave the negative terms in their original form, we shall have

0=(x−1).x11+(x−1 )x2¬2+(x−1).x2¬3+...+x−1+1
+P(-1)+P1(x−1).x"¬3+...+P1(x-1)+Pi

+P2(x-1)x13+...+P2(x-1)+P2

+

Now if such a value be assigned to r that every term is positive, that value will be the superior limit required; in the terms where no negative coefficient enters, it is sufficient to have x>1; in the other terms, each of which involves a negative coefficient, we must have

If, then, r be taken equal to the greatest of these fractions increased by unity, this value, and every greater value, will make ƒ(x) positive, and therefore will be a superior limit of the positive roots. This method gives a limit easily calculated, and generally not far from the truth.*

(2) 4x7—6x6—7x2+8x2+7x3—23x2—22x−5=0;

here 3 is a superior limit.

OBSERVATION.-The form of the equation will often suggest artifices, by means of which closer limits may be determined than by any of the preceding methods; thus, writing the equation of Example 1 under the form

4x^(x-2)+23x2+105x(x—19)+3=0,

we see that x= or >2 gives a positive result, therefore 2 is a superior limit. Similarly, by writing the example of Art. 284 under the form

z(z−25)+1(n)=

we see that 3 is a superior limit.

We have seen (Art. 248) that an equation of an even number of dimensions with its last term positive may have no real root; but we shall now show that

This is the method of Bret.

in any equation whatever, if the absolute term be small compared with the other terms, there will be at least one real root also very

286. In the equation

Porn +P12-1+, &c., +x−r=0,

small.

where r is essentially positive, and which may represent any equation what

ever, if r<

1

'4(1+p)'

where p is numerically the greatest coefficient, then there is a real positive root, <2r.

By dividing by the coefficient of x, and changing the signs of all the terms, and of all the roots, if necessary, every equation may be reduced to the form −r+x+, &c., +P1x"1+Pox"=0 .... (1)

where r is essentially positive; let p be numerically the greatest coefficient, then any value of <1 which makes

will make the first member of (1) positive; and this condition is fulfilled by

2(1+p)x=(1+r)— √(1+r)2 —4r(1+p);

if, then, 4r(1+p)<1, the radical will have a real value >r, and there will be

1

for r a real value less than

2(1+P)

which makes the first member of (1) positive, while x=0 makes it negative; therefore, in any equation reduced to the

1

above form, if r<

there is a real small positive root, <2r.

·4(1+P)

EXAMPLE.

x2+18x3—21x2—12x+1=0

has a real root between 0 and

6'

287. To find an inferior limit of the positive roots, we must transform the equation into one whose roots are the reciprocals of the roots of the former; and the reciprocal of the superior limit of the roots of the transformed equation, found by the preceding methods, will be the quantity required. Hence, if p, denote the greatest coefficient of a contrary sign to the last

term, p1, an inferior limit of the positive roots is

equation will be (Art. 274)

is the greatest negative coefficient; therefore +1 is a superior

Pr

Pn limit of its roots; and, consequently, of the proposed equation.

Pa

ph Pr+Pa

an inferior limit of the positive roots

which evidently has 9 for the superior limit of its positive roots, and, there

1

fore, the proposed has for its inferior limit.

288. To find superior and inferior limits of the negative roots, we must transform the equation into one whose roots are those of the former with contrary signs (Art. 247); and if a, ẞ be limits, found as above, of the positive roots of this equation, then -a and ẞ will be limits of the negative roots of the proposed equation.

EXAMPLE.

x3-7x+7=0;

putting x=-y, we get y3-7y-7=0, of which 1+ √7 or 4 is a superior limit.

3

is a superior limit; therefore the negative root of the proposed lies between -4 and -3.

NEWTON'S METHOD OF FINDING LIMITS OF THE ROOTS.

289. The limits, however, deduced by any of the preceding methods seldom approach very near to the roots; the tentative method, depending upon the following proposition, will furnish us with limits which lie much nearer to them.

Every number which, written for x, makes f(x) and all its derived functions positive, is a superior limit of the positive roots.

For, if we diminish the roots a, b, c, &c., of ƒ (x)=0 by h, that is (Art. 251), substitute y+h for x, the result is f(y+h)=0, or

f(h)+f'(h)}{+ƒ''(h);

y2

1.2

+

...

+f()_+3=0.

Now, if we give such a value to h that all the coefficients of this equation are positive, then every value of y is negative; that is, all the quantities, a—h, b―h, c—h, &c., are negative, and therefore h is greater than the greatest of the quantities a, b, c, &c., or is a superior limit of the roots of the proposed equation. Similarly, h will be an inferior limit to all the roots, if the coefficients be alternately positive and negative.

EXAMPLE.

To find a superior limit of the roots of

x3-5x2+7x-1=0.

The transformed equation, putting y+h for x, is

(h3 —5h2+7h−1)+(3h2—10h+7)y+(6h—10)31⁄2+y3=0;

in which, if 3 be put for h, all the coefficients are positive; therefore 3 is a superior limit of the positive roots.

OBSERVATION. This method of finding a superior limit of the roots by determining by trial what value of x will make ƒ(x) and all its derived functions positive, was proposed by Newton.

WARING'S OR LAGRANGE'S METHOD OF SEPARATING THE ROOTS.

290. If a series of quantities be substituted for x in f(x), then between every two which give results with different signs an odd number of roots of f(x)=0 is situated; and between every two which give results with the same sign an even number is situated, or none at all; but we can not assure ourselves that in the former case the number does not exceed unity, or that in the latter it is zero, and that, consequently, the number and situation of all the real roots is ascertained, unless the difference between the quantities successively substituted be less than the least difference between the roots of the proposed equation; since, if it were greater, it is evident that more than one root might be intercepted by two of the quantities giving results with different signs, and that two roots instead of none might be intercepted by two of the quantities giving results with the same sign, and in both cases roots would pass undiscovered. We must, therefore, first find a limit less than the least difference of the roots; this may be done by transforming the equation into one whose roots are the squares of the differences of the roots of the proposed equation. Then, if we find a limit k less than the least positive root of the transformed equation, √ will be less than the least difference of the roots of the proposed equation; and if we substitute successively for r the numbers s, s— s—2 √k, &c. (s being a superior limit of the roots of the proposed), till we come to a superior limit of the negative roots, we are sure that no two real roots lying between the numbers substituted have escaped us, and that every change of signs in the results of the substitutions indicates only one real root. Hence the number of real roots will be known (for it will exactly equal the number of changes), as well as the interval in which each of them is contained.

OBSERVATION. This method of determining the number and situation of the real roots of an equation was first proposed by Waring; it is, however, of no practical use for equations of a degree exceeding the fourth, on account of the great labor of forming the equation of differences for equations of a higher order.

EXAMPLE.

x3-7x+7=0.

The numbers 1 and 2 give each a positive result, but yet two roots lie between them. The equation whose roots are the squares of the differences is (Art. 279) y3-42y+441y-49=0, an inferior limit of the positive roots of which is (Art. 287); therefore, is less than the least difference of the

1 9

1 3

hence, one value of x lies between 2 and and one between and 3

; and,

similarly, we find the negative root, which necessarily exists, to lie between -3

5 3'

5

4

METHOD OF DIVISORS.

291. The commensurable roots of f(x)=0, which are necessarily whole numbers, may be always found by the following process, called the method of divisors, proposed by Newton.

Suppose a to be an integral root; then, substituting a for x, and reversing the order of the terms, we have

Pa

Hence, is an integer which we may denote by q1; substituting and di

a

a

Similarly, is an integer =92 suppose; and proceeding in this manner, we shall at last arrive at

1

Hence, that a may be a root of the equation, the last term, pa, must be divisible by it, so must the sum of the quotient and next coefficient, q1+Pa−1; and continuing the uniform operation, the sum of each coefficient and the preceding quotient must be divisible by a, the final result being always 1.

If, therefore, we take the quotients of the division of the last term by each of the divisors of the last term which are comprised within the limits of the roots, and add these quotients to the coefficient of the last term but one; divide these sums, some of which may be equal to zero, by the respective divisors, add the new quotients which are integers or zero (neglecting the others) to the next coefficient and divide by the respective divisors, and so on through all the coefficients (dropping every divisor as soon as it gives a fractional quotient), those divisors of the last term which give -1 for a final result are the integral roots of the equation; and we shall thus obtain all the integral roots, unless the equation have equal roots, the test of which will be that some of the roots already found satisfy f'(x)=0, and the number of times that any one is repeated will be expressed by the degree of derivation of the first of the derived functions which that root does not reduce to zero, when written in it for x (Art. 253). It is best to ascertain by direct substitution whether +1 and 1 are roots, and so to exclude them from the divisors to be tried.

EXAMPLE I.

3+3x-8x+10=0.

8

Here the roots lie between 4+1 and 11 (Arts. 285, 288), and the divisors of the last term are ±{2, 5, 10},