Its characteristic is, that when it, or any number greater than it, is substituted for x in the equation, the result is positive.

An Inferior Limit to the negative roots, is a number numerically greater than the greatest negative root. The substitution of it, or any number greater than it, for x, produces a negative result.

The object of ascertaining the limits of the roots is to diminish the labor necessary in finding them.

416. Proposition I.-The greatest negative coëfficient, increased by unity, is greater than the greatest root of the equation.

and suppose A to be the greatest negative coëfficient. The reasoning will not be affected if we suppose all the coefficients to be negative, and each equal to A.

It is required to find what number substituted for x will make x">A(x1+x2¬2+xn−3.... +x+1).

By Art. 297, the sum in parenthesis is

; hence, we must

X

By considering all the coefficients after the first negative, we have taken the most unfavorable case; if either of them, as B, were positive, the quantity in parenthesis would be less.

417. Proposition II.-If we take the greatest negative coefficient, extract a root of it whose index is equal to the number of terms preceding the first negative term, and increase it by unity, the result will be greater than the greatest positive root of the equation.

Let Can be the first negative term, C being the greatest negative coëfficient; then, any value of x which makes

x">С(x"¬"+xn--1.....+x+1) (1)

will evidently render the first member of the inequality >0, or positive; because this supposes all the coëfficients after C negative, and each equal to the greatest, which is evidently the most unfavorable case.

or, by multiplying both members by x-1, and dividing by "-+1,

But x-1 is <x, and .. (x−1)r-1<x3-1 .. (2) will be true if we have (x-1)(x-1)-1,

Or (x-1)=C, or >C;

Or x-1=VC, or >VC;

Or x=1+VC, or >1+VC.

Find superior limits of the roots of the following equa

tions:

1. x1—5x3+37x2-3x+39=0.

Here, C=5, and r=1 .. 1+†/0=1+51—6, Ans.

2. x+7x-12x3-49x2+52x-13-0.

Here, 1+0=1+2/49=1+7=8, Ans.

3. x+11x2-25x-67-0.

By supposing the second term +0x3, we have r=3; hence, the limit is 1+/67, or 6.

418. To determine the inferior limit to the negative roots, change the signs of the alternate terms; this will change the signs of the roots (Art. 400); then,

The superior limit of the roots of this equation, by changing its sign, will be the inferior limit of the roots of the proposed equation.

419. Proposition III.—If the real roots of an equation, taken in the order of their magnitudes, be a, b, c, d, etc., a being greater than b, b greater than c, and so on; then, if a series of numbers, a', b', c', d', etc., in which a' is greater than a, b' a number between a and b, c' a number between b and c, and so on, be substituted for x in the proposed equation, the results will be alternately positive and negative.

The first member of the proposed equation is equivalent to (x—a)(x—b)(x—c)(x—d). . . =0.

....

Substituting for x the proposed series of numbers a', b', c', etc., we obtain the following results:

(a′—a) (a'—b)(a’—c)(a’—d), etc. the factors are +.

(b’—a) (b'—b)(b’—c)(b′—d), etc. one factor is

(c'—a) (c'—b) (c'—c)(c'—d), etc.

factors are, and the rest +. (d'-a)(d'-b)(d'—c) (d'-d), etc..

odd number of factors is

+ product, since all

product, since only

+ product, since two

and so on.

- product, since an

Corollary 1.—If two numbers be successively substituted for x, in any equation, and give results with contrary signs, there must be one, three, five, or some odd number of roots between these numbers.

Corollary 2.-If two numbers, substituted for x, give results with the same sign, then between these numbers there must be two, four, or some even number of real roots, or no roots at all.

Corollary 3.-If a quantity q, and every quantity greater than q, render the results continually positive, q is greater than the greatest root of the equation.

Corollary 4.-Hence, if the signs of the alternate terms be changed, and if p, and every quantity greater than p, renders the result positive, then -p is less than the least root of the equation.

ILLUSTRATION. If we form the equation whose roots are 5, 2, and -3, the result is x3-4x2-11x+30=0. Now, if we substitute any number whatever for x, greater than 5, the result is positive. If we put x=5, the result is zero, as it should be.

If we substitute for x, any number less than 5, and greater than 2, the result is negative. Putting x=2, the result is zero.

Substituting for x, any number less than 2, and greater than -3, the result is positive. Substituting -3, it is zero.

Substituting a number less than -3, the result is negative.

From Cors. 3 and 4, it is easy to find when we have passed all the real roots, either in the ascending or descendiug scale.

STURM'S THEOREM.

420. To find the number of real and imaginary roots of an equation.

In 1834, M. Sturm gained the mathematical prize of the French Academy of Sciences, by the discovery of a beautiful theorem, by means of which the number and situation of all the real roots of an equation can, with certainty, be determined. This theorem we shall now proceed to explain.

-1

Let X=x+Ax-1+Вx-2. +Tx+V=0, be any equation of the nth degree, containing no equal roots; for if the given equation contains equal roots, these may be found (Art. 414), and its degree diminished by division.

Let the first derived function of X (Art. 411) be denoted by X1.

Divide X by X1 until the remainder is of a lower degree with respect to X than the divisor, and call this remainder X2; that is, let the remainder,

with its sign changed, be denoted by X2.

X1)X (Q1
X1Q1

X-X1Q1--X2

X2) X1 (Q2
X2Q2
X1-X2Q2=-X3

X3) X2 (Q3
X3Q3

Divide X by X2 in the same manner, and so on, as in the margin, denoting the successive remainders, with their signs changed, by X3, X4, etc., until we arrive at a remainder which does not contain x, which must always happen, since the equation having no equal roots, there can be no factor containing x, common to the equation and its first derived function. Let this remainder, having its sign changed, be called Xr+1•

X2-X3Q3=-X4

In these divisions, we may, to avoid fractions, either multiply or divide the dividends and divisors by any positive number, as this will not affect the signs of the functions X, X1, X2, etc.

By this operation, we obtain the series of quantities

Each member of this series is of a lower degree with respect to x than the preceding, and the last does not contain x. Call X the primitive function, and X1, X2, etc., auxiliary functions.

421. Lemma I.--Two consecutive functions, X1, X2, for example, can not both vanish for the same value of x.

From the process by which X1, X2, etc., are obtained, we have the following equations:

If possible, let X=0, and X2=0; then, by eq. (2) we have X=0; hence, by eq. (3) we have X=0; and proceeding in the same way, we shall find X-0, X=0, and finally Xr+1=0. But this is impossible, since X,+1 does not contain x, and therefore can not vanish for any value of x.