Number of Real Roots.

when x∞ it is

+(unlike a);

so that there is no real root when a is positive, and two real roots when a is negative, which agrees with art. 285.

is

Second case. If n is odd, the row of signs when x = ∞

when x

+,+, = (unlike a);

∞ it is

-,, (unlike a);

so that, in either case, there is only one real root, which is, by art. 284, of a sign unlike that of a.

4. Find the number of real roots of the equation

First case. When n is even and greater than 2, and U" positive, that is, when b is positive, and

so that when a is positive, there is no real root, and when a is negative there are two real roots. In the latter case, the row of signs when x = 0 is

Number of Real Roots.

Second case. When n is even and greater than 2, and U" zero, that is when b is positive and

so that in either case there is no other real root than the

above equal root.

Third case. When n is even and greater than 2, and U" negative, that is,

so that when a is negative there is no real root, and when a is positive there are two real roots. In the latter case, the row of signs when x = 0 is

(like b), +, ‡ (unlike b), -;

so that when b is positive, both the roots are negative, and when b is negative, one of the roots is positive and the other negative, which agrees with art. 287.

Fourth case. When n is odd, and U" positive, that is, when a is negative and

Number of Real Roots.

in which case, the row of signs when x = ∞ is

so that the equation has three real roots; the row of signs when x = 0 is

so that when b is negative, one of the real roots is positive and the other two negative; and when b 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

so that there is another real root besides the above equal The row of signs when x = 0 is

root.

(like b),, (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,

(一)<(__)

Number of Real Roots.

in which case, the row of signs when x = ∞ is

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 x3 — 6 x2 + 19 x — 44 0.

Ans. It has one positive real root.

5. Find the number of real roots of the equation

Ans. It has four positive real roots.

6. Find the number of real roots of the equation 0.

x1 + x3 24 x2 + 43 x 21=

Ans. It has three positive real roots and one negative

7. Find the number of real roots of the equation

one.

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

by u and its first, second, &c. derivatives by U, U', &c.

of

Stern's Theorem.

Find the row of signs corresponding to the values

u, U, U', U", &c.,

for any value p of the variable, and also for a value q 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 x decreases from a value p 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 Ρ and q.