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ranging the results according to the ascending powers of u.
Observe that X, is the first derived polynomial of X,; hence, adopting the same notation as in (436), we have, from the two equations,
- 0, (1)
X , ;
(2) 2 2:3 where, it will be observed, X', X1, X'2, etc., represent what X, X1, X,, etc., become, when x takes the place of x. Now suppose
x' =r; that is, x = rtu, r being any root of the given equation. Then X = 0; and as X'ı, X',, X'z, now receive definite values, the values of X and X, may, or may not become zero by giving a particular value to u. Dropping Y' from (1), and factoring the result, we have
where the different terms may be essentially positive or negative, according to the values or r and u, upon which they depend.
Now, it is evident that by causing u to diminish numerically, each term after the first, in the parenthesis, may be made as small as we please; and by making u sufficiently small, the sum of the terms containing u, in each parenthesis, may be made less than the first term X" 1; in which case the essential sign of the quantity in either parenthesis will depend upon the sign of X'',. Thus, when u is indefinitely small, the signs of the functions, X and X,, will depend upon the signs of u (X',) and X',, respectively. Hence, when u is negative, X and X, will have opposite signs; but when u is positive, X and X, will have the same signs.
457. Thus we have shown, that if we substitute in a given equation X = 0, and its first derived polynomial X, = 0, a quantity r—u, which is insensibly less than the root r, the results will have opposite signs ; but if we substitute the quantity r+u, which is insensibly greater than the root r, the results will have the same sign.
458. Consider the quantity substituted in the two functions to be insensibly less than a, the least root of X= 0, and let it increase
X = a-U
till it is insensibly greater than a. In passing the root a, the function X will change sign, (452,3); hence the signs of the functions will be as follows:
X X, +
or else 0 + x = atu
+ + Now let the substituted quantity increase fromx = atu to
56—u, a value insensibly near to b, the next root of X= 0. According to the principle already established (457), X and X, must now have opposite signs. And since X can not have changed its sign during the change of x from a+u to 1—u, there must have been a change of sign in the function X,. Hence, by (452,3) one root of X 0 is found between a tu and b-u, or between a and y. In like manner, it can be shown that X, 0 has one root between b and c, one between c and d, and so on. osition is proved.
Hence the prop
459. The object of Sturm's Theorem is to determine the number of the real roots of an equation, and likewise the places of these roots, or their initial figures when the roots are irrational.
NOTE.-- This difficult problem, which for a long time baffled the skill of mathematicians, was first solved by M. Sturm, his solution being submitted to the French Academy in 1829.
460. We have seen, (435), that the equal roots of an equation may always be found and suppressed. Now let
X = + Amm-1+B.*-?+ ....Tx+u=0 represent any equation having no equal roots, and X, = 0 its first derived polynomial, or its limiting equation. We will now apply to the functions, X and X1, a process
similar to that required for finding their greatest common divisor (105), but with this modification, namely; that we change the signs of the successive remainders, and neither introduce nor reject a negative factor, in preparing for division.
Denote the successive remainders, with their signs changed, by
R, R1, R2,....R-1, Rn. Since the given equation has no equal roots, there can be no common divisor between X and X1, (435); hence, if the process of division be continued sufficiently far, the last remainder, Rn, must be different from zero, and independent of x.
Now in the several functions, X, X, R, R1, R2,....R-1, RM, let us substitute for x any number, as h, and having arranged the signs of the results in a row, note the number of variations of signs. Next substitute for x a number, h', greater than h, and again note the number of variations of signs. The difference in the number of variations of signs, resulting from the two substitutions, will be equal to the number of real roots comprised between h and h'.
This is Sturm's Theorem, which we will now demonstrate.
Let Q, Q1, Q2,.... Qx-1, In denote the quotients in the successive divisions. Now in every case, the dividend will be equal to the product of the divisor and quotient, plus the true remainder, or minus the remainder with its sign changed. Hence, (1)
X = Xl R (2)
X = R Q - R (3) R = RQ2
R, = R2Q3
- (n) R-2 = R-On From these equations, it follows,
1-If any number be substituted for x in the functions X, X,, R, R1,.... Rn, no two of them can become zero at the same time.
For, if possible, let such a value of h be substituted for x as will render X, and R zero at the same time. Then the second equation of (4) will give Ri=0; whence, the third equation will become R=0; and tracing the series through, we shall have, finally, R,= 0, which is impossible.
2.- If any one of the functions become zero by substituting a particular value for x, the adjacent functions will have contrary signs for the same value.
For, suppose R, in the third equation to become zero; then this equation will reduce to R :-R2. That is, R and R2
have contrary signs.
Having established these principles, suppose the quantity h, which
is to be substituted simultaneously in all the functions, to be a variable, changing by insensible degrees from a less to a greater value. As it passes any of the roots, the function to which this root belongs will reduce to zero, and change sign, (452, 3). Let р be a little less than a certain root of R2, 9
little greater than the same root, the two values being so taken, however, that no root of R, or R, shall be comprised between them. As h changes from p to 9, R2 will reduce to zero, and change sign. But neither R, nor R, will change sign; and since, according to the second principle, these functions have opposite signs when R2 0, they must have opposite signs also when h =p or h =q. Now when h EP, the arrangement of signs must be
R, R2 R3 R, R2 R3
+ +; giving one variation and one permanence, whichever way the double sign be taken. When h
the signs must become R, R2 R3 R, R, R3
+ F giving, as before, one variation and one permanence, so that the whole number of variations is neither increased nor diminished.
This reasoning obviously applies to any function which is situated between two other functions.
Hence, 3.— When h passes a root of any function intermediate between X and Rn, the number of variations of signs will not be altered.
As the last function, Rn, is independent of x, its sign will not be changed by any substitution for x. It follows, therefore, that if any change is produced in the number of variations of signs, it must result from the alternation of signs in the original function X.
Let a, b, c, d, ....l be the roots of X, taken in the order of their values. Then the roots of X, will be found, the first between a and b, the second between b and c, and so on; (458). The degree of X, is less by 1 than the degree of X; hence, if the degree of X is odd the degree of X, will be even, and if the degree of X is even the degree of X, will be odd. Now take h less than a; according to (452, 1), the signs of X and X, will be unlike, giving a variation. Let h increase till it is insensibly greater than a; X will change sign, and the variation between X and X, will be lost.
Now let h increase till it is insensibly less than b. It will pass the first root of X1, causing the signs of X and X, to be again unlike; but by (3), this change in the sign of X, will not alter the whole number of variations in the signs of the functions. Again let h increase till it is insensibly greater than b; X will again change sign, and another variation will be lost. In like manner it may be shown that the number of variations will be diminished by 1 every time h pal: ses a root of X; hence the truth of the theorem.
461. If we substitute for x in the several functions h = and h' + co, successively, we shall determine at once the whole number of real roots in the given equation. To ascertain the signs of the functions resulting from these substitutions, we require the following principle :
If in any polynomial involving the descending powers of x, infinity be substituted for x, the sign of the whole expression will depend upon the sign of the first term.
Let Axm + Bxm-' + Cwm-?+Dxm-s +Exm-sto... be the given polynomial. If x= 00, then
B C D
(1) oc? 23 because every term in the second member is less than any assignable quantity, or zero, (188, 2). Multiplying both members of (1) by x”, we have
Ax" > Bx*-' + 0.0"-+Dxm-s+ Ex-+.... That is, when = 0, the first term of the given polynomial is numerically greater than the sum of all the other terms. Hence the sign of the whole will be the same as the sign of the first term.
462. In the application of Sturm's Theorem, we may always suppress any numerical factor in any of the functions X, R, R, etc.; for this will not affect the sign of the result.
1. Given the equation _3x*–12.c+24=0, to find the num ber and situation of the real roots. Suppressing monomial factors, we have for the several functions,
12x + 24,
2, R 4.