Number of Real Roots of an Equation of an Odd Degree.

284. Theorem. Every equation of an uneven degree, has at least one real root affected with a sign contrary to that of its last term, and the number of all its roots is uneven.

Proof. Let the equation be

x2 + a x2-1+ &c. ... +m=0,

the first term of which is infinitely greater than any other term, or than the sum of all the other terms.

The sign

of this result is therefore the same as that of its first term,

which may be shown by the above reasoning to be negative. The given equation must then, by art. 281, have at least one real root, and by art. 283, the number of its real roots must be uneven.

Secondly. To prove that one, at least, of the real roots is affected with a contrary sign to that of the last term. The substitution of

x = 0,

reduces the given first member to its last term m.

Number of Real Roots of Equations.

Comparing this with the above results, we see that, if m is positive, the given equation must, by art. 281, have a real root contained between 0 and ∞, that is, a negative root; but if m is negative, there must be a real root contained between 0 and +∞, that is, a positive root; so that there must always be a root affected with a sign contrary to that of m.

285. Theorem. The number of real roots if there are any, of an equation of an even degree must be even, and if the last term is negative, there must be at least two real roots, one positive and the . other negative.

Proof. Let the equation be

x2 + a x2-1+bxn−2+ &c. .

in which n is even.

+m=0,

First. To prove that the number of real roots is even.

gives for the value of the first member

∞2 + a con-1+b∞n-2+ &c. .

...

+m,

Hence, if the given equation has any real root, there must, by art. 283, be an even number of them.

Secondly. To prove that when m is negative, there must be two real roots, the one positive, the other negative. The substitution of

x=0

Number of Imaginary Roots; of Real Positive Roots.

reduces the given first member to its last term m, and this result is therefore negative in the present case.

Comparing this with the above results, we see that there must be a real root between 0 and ∞, and also one between 0 and ; that is, the given equation has two real roots, the one positive and the other negative.

286. Corollary. Since the number of real roots of an equation of an uneven degree is uneven, and that of an equation of an even degree is even, the number of imaginary roots of every equation, which has imaginary roots, must be even.

287. Theorem. The number of real positive roots of an equation is even, when its last term is positive; and it is uneven, when the last term is negative.

gives, for the first member of the given equation, a positive result; while the substitution of

x= 0

reduces the first member to its last term.

Hence if this last term is positive, the number of real roots contained between 0 and ∞, that is, of positive roots, must, by art. 283, be even; and if this last term is negative, the number of these roots must be uneven.

288. Theorem. If a function vanishes, that is, is equal to zero for a given value x' of its variable x; the function and its derivative must have like signs for a value of the variable which exceeds x' by

Variation and Permanence.

an infinitely small quantity, and unlike signs for a value of the variable which is less than x' by an infinitely small quantity.

Proof. Let the given function be u, and its derivate U, and, as in art. 176, when the variable is increased by the infinitesimal i, the function becomes

u + U i.

This value of the function, when

u = 0

is reduced to Ui, which has, obviously, the same sign with U.

In the same way when the variable is decreased by i, the function becomes

is reduced to Ui, having the opposite sign to U.

289. Definition. A pair of two`successive signs, in a row of signs, is called a permanence when the two signs are alike, and a variation when they are unlike.

290. Sturm's Theorem. Denote the first member of the equation

x2 + a x2-1+&c.

by u and its derivative by U.

common divisor of u and U, and,

= 0

Find the greatest

in performing this

process, let the several remainders which are of continually decreasing dimensions in regard to x, be denoted, after reversing their signs, by

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

Sturm's Theorem.

Find the row of signs corresponding to the values

of

for any value p

9

of the variable.

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

of the variable, and also for a value

The difference between the number of permanences of the first row of signs, and that of the second, is exactly equal to the number of real roots of the given equation comprised between p and q.

Proof. The method in which U', U", &c., are obtained gives, at once, by denoting the successive quotients in the process by m, m', &c.

First. Two successive terms of the series cannot vanish at the same time, except for a value of x which is one of the equal roots of the given equation. For when U" and U", for instance, are zero, the equation

so that the function and the derivative are both zero at the same time, which, by art. 278, corresponds to the case of one of the equal roots of the equation.

Secondly. If any term of the series, except the first or last, has a different sign in the row corresponding to the value P of the variable from that which it has in the row