When the numbers substituted differ by unity, it is evident that the integral part of the root is known.

EXAMPLES.

1. What is the first figure of a root of the equation x3-3x2-7x-8=0?

Here, x2 gives X-2, and x=3 gives X=25; hence 2 is the first figure of the root.

2. Find the integral parts of all the roots of the equation x3 - 6 x2 + 3 x + 9 = 0.

475. If any quantity numerically greater than either a or (Art. 473) be substituted for x, each binomial factor of (B) will take the sign of x, and the sign of X will be the same as that of x2.

If any quantity numerically less than each root of X=0 be substituted for x, each binomial factor will take the sign of its last term, and the sign of X will be the same as that of v, its absolute term (Art. 441).

STURM'S THEOREM.

476. To determine the number and situation of the real, roots of any equation.

A perfect solution of this difficult problem was first obtained by Sturm, in 1829. As the theorem which bears his name determines the number of real roots, the number of imaginary roots also becomes known (Art. 436). 477. Let X represent the first member of the equation

from which the equal roots have been removed (Art. 471), and X, the first derived polynomial of X (Art. 467).

Divide X by X1, and we shall obtain a quotient, Q, with a remainder of a lower degree than X1. Represent this remainder, with its signs changed, by X2, divide X1 by X2, and so on, as in finding the greatest common divisor (Art. 108), except that the signs of the remainders must be changed, while no other change of signs is admissible. As the equation X=0 has no equal roots, there can be no common divisor, and the last remainder, X, will be independent of x. The successive operations may be represented by the following equations:

478. Substitute any number for x in the series of functions represented by X, X1, X2,

the signs of the results in a row. be substituted in like manner,

.... X, and write If any other number

The difference between the number of variations in the first row of signs and that in the second, is equal to the number of real roots of the given equation comprised between the quantities substituted for x.

This is Sturm's Theorem, the demonstration of which depends upon the following principles.

1. Two consecutive functions of the series X, X1, X2, .... Xn cannot both become 0 for the same value of x.

If X=0 and X=0, then, by equation (2), Art. 477, X 0; and if X = 0 and X, 0, then, by (3), = X=0; and so on, till X11 = 0. But it is impossible that the last remainder shall become 0, for any value of x, because it is independent of x.

2. When either function of the series, except X and X:

becomes 0 for a particular value of x, the two functions adjacent to it must have different signs.

If X2: 0 when x=b, then neither X, nor X, can be 0 for that value of x, and, by equation (2), X1 = that is, X and X must have different signs.

It is evident that X and X, will also continue to have different signs for values of x a little greater or less than b; for no change in the signs of X1 or X, can take place until after the value of x is so much changed that one of them becomes 0 (Art. 473).

3. When either function of the series, except X and X, changes its sign for different values of x, the NUMBER of variations is not affected by the change.

If c and d, when substituted for x, give different signs for X, conceive all possible quantities between c and d to be substituted for x, in regular order, and X, changes its sign in passing through 0 (Art. 473). Hence, when X, changes its sign, as well as just before and just after the change, X and X, must have different signs, and the sign of X2, whether or must, therefore, in both cases, be like one of those adjacent to it, and unlike the other, giving, with X and X, one permanence and one variation. The only effect of changing the sign is that the permanence and variation exchange places.

The last function, X, is independent of x, and therefore can never change its sign, whatever value may be given to x. Hence a change in the number of variations can be caused only by a change in the sign of the original function, X.

4. When the original function, X, changes its sign for successive increasing values of x, the number of variations is diminished by one.

Let a be a root of the original equation, or a value of x which causes X to become 0, and let A1 represent the

1

If we

value of X1 when x = a. assume x = a + y and xa — y, y being too small for X1 to become 0, then the sign of X1 for each of these values of x will be the same as for xa, that is, the same as the sign of A1.

But if ay be substituted for x in X, we obtain (Art. 469)

or, since A is merely a substituted for x in X, and therefore equal to 0,

As these series are finite, y may be made so small that the sign of each shall depend upon that of A, (Art. 475). Hence, X has the same sign as A when x is a little larger than a, and a different sign when it is a little smaller. But X has the same sign as A. Hence, if x is increasing, X and X1 have a different sign before X becomes 0, and the same sign after.. A variation is therefore changed to a permanence.

The truth of the theorem is evident from the foregoing principles. If x take all possible values from p to 9, p being less than q, one variation is changed to a permanence each time that X passes through 0 and changes its sign, and only then, for no change of sign in any of the other functions, X, X2, etc., can affect the number of variations. Hence the number of variations lost in passing from p to q is exactly equal to the number of real roots of the equation X=0 comprised between p

and q.

479. When -∞ and

∞ are substituted for Ρ and q, the whole number of real roots of the equation X=0 becomes known (Art. 473).

The substitution of -∞ and 0 will give the whole number of negative roots, while 0 and ∞ will give the whole number of positive roots. When the roots are all real, Descartes' Rule (Art. 452) will answer the same purpose.

The substitution of various numbers for Ρ and q will show between what numbers the roots lie, or fix the limits of the roots.

480. It is evident that X and X, must change signs alternately, as they are always unlike just before X changes. Hence, when the roots of X=0 are all real, each root of X1 = 0 must be intermediate in value between two roots of X = 0. For this reason the first derived equation is often called the limiting or separating, equation (Art. 468).

481. In the process of finding X,, X, etc., any positive numerical factors can be introduced or rejected at pleasure, since the sign of the result is not affected thereby. In this way fractions may be avoided.

In substituting

∞ and∞, the first term of each function determines the sign (Art. 475).

1.

EXAMPLES.

Determine the number and situation of the real roots

4x2

of the equation x34x2-x+4=0.

Here, the first derived polynomial of the first member (Art. 467) is 3x2 - 8 x — 1. Multiplying 3 — 4 x2 — x 4 by 3, to make its first term divisible by 3x2,