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Number of Real Roots.
co it is
in which case, the row of signs when a = co is
tot, tet; when x =
-, +,-, ti 80 that the equation has three real roots ; the row of signs when x=0 is
+(like b), -, F (unlike b), t; so that when 6 is negative, one of the real roots is positive and the other two negative ; and when 6 is positive, one of the real roots is negative and the other two positive.
Fifth case. When n is odd, and U" zero, that is, when a is negative and
on it is
in which case, there is the equal root
(n-1) a the row of signs when x = o is
to tt; when x=
-, +,-; so that there is another real root besides the above equal root. The row of signs when x = 0 is
+ (like b), -, F (unlike b); so that one of the roots is positive and the other negative.
Sixth case. When n is odd, and U" negative, that is,
Number of Real Roots.
in which case, the row of signs when w=w is
+ + F (unlike a), - ; when x = -00 it is
-t, (like a), so that there is only one real root, which, by art. 284, has a sign contrary to that of its last term. 4. Find the number of real roots of the equation
23 – 6 x2 + 19 2 — 44 = 0.
Ans. It has one positive real root. 5. Find the number of real roots of the equation
10 23 + 35 x2 50 x + 24 = 0.
Ans. It has four positive real roots. 6. Find the number of real roots of the equation
24 + 203 — 24 22 + 43 x 21 = 0. Ans. It has three positive real roots and one negative
7. Find the number of real roots of the equation
24 + 8x2 + 16 x 440 = 0.
Ans. One positive root and one negative root. 293. Sturm's theorem is perfect in always giving the number of real roots, but often requires so much labor, that theorems, which are much less perfect, may be used with great advantage.
294. Stern's Theorem. Denote the first member of the equation
2n + a x + &c. = 0 by u and its first second, &c. derivatives by U, U', &c.
Find the row of signs corresponding to the values of
U, U, U', U", &c., for any value
р of the variable, and also for a value 9 of the variable.
The number of real roots of the equation, comprised between p and q, cannot be greater than the difference between the number of permanences of the first row of signs and that of the second row.
Proof. First. It may be shown as in the third division of the proof of art. 290, that one permanence at least is always lost from the row of signs when the variable in decreasing passes through a value which is one of the roots of the equation.
Secondly. When any term of the series except the first or the last, vanishes, it passes, by art. 288, with the decreasing variable, from having the same sign with its derivative, which is the next term of the series, to having the reverse sign of it. Even then, if it had before the change the reverse sign of the preceding term and after the change the same sign, it introduces a permanence which is only sufficient to take the place of the other permanence which is lost. The number of permanences of the row of signs is not, therefore, augmented by the vanishing of such an intermediate term.
Thirdly. The last term of the series must be constant, for the number of dimensions is diminished by each derivation; and, therefore, as a decreases from a value р to a smaller value q, the number of permanences of the row of signs must be diminished by as large a number at least as the number of roots comprised between p and q.
295. Definition. An equation
20tar-1 + &c. + k 22 + k x+1=0. is said to be complete in its form, when it contains terms multiplied by every different power of x from the highest to unity, and also a constant term, such
296. Descartes' Theorem. A complete equation cannot have a greater number of positive roots than there are variations in the row of signs of its terms, nor a greater number of positive roots than there are permanences in this row of signs.
Proof. If the equation is that of art. 295, the values of , U, U', &c. in art. 294, are u = 2n tax-1+ &c. + h x2 + kuti U=nrn-1 + (n − 1) arn–2 +&c. +2hz+k U'= n(n-1) 2-3+(n-1)(n-2)az*=36&c.+2h
&c. The row of signs when x = 0 is
tit, te tot, &c., consisting wholly of permanences. When x=- it is
+, F, E, F, &c., in which the upper row of signs is used when n is even, and the lower row when n is odd.
this consists wholly of variations. The row of signs when x = 0 is
+ (like 2), + (like k), + (like 2 h) that is, it is the same as the row of signs formed by the terms of the equation taken in the inverse order.
In either case,
Zero substituted for a Term which is wanting.
The limit of the number of positive roots is, therefore, by art. 294, equal to the excess of the whole number of pairs of successive signs of the terms, over the number of permanences ; that is, it is equal to the number of variations ; and in the same way, the number of negative roots cannot exceed the number of permanences.
297. Corollary. The whole number of successions of signs of an equation, that is, the sum of the permanences and variations, is one less than the number of terms, or the same as the degree of the equation, that is, the same as the number of roots.
If, therefore, all the roots are real, the number of positive roots must be the same as the number of variations, and the number of negative roots must be the same as the number of permanences.
298. Scholium. Whenever a term is wanting in an equation, its place may be supplied by zero, and either sign may be prefixed.
299. Corollary. When the substitution of +0 for a term which is wanting gives a different number of permanences from that which is obtained by the substitution of — 0, and consequently a different number of variations also, the equation must have imaginary roots.
300. Theorem. When the sign of the term which precedes a deficient term is the same with that which follows it, the equation must have imaginary roots.
Proof. For if the terms which precede and follow the deficient term are both positive, the substitution of +0 gives two permanences; while the substitution of