Number of Real Roots between two given Numbers. and de value of Y by Y'; and when we substitute q for x, note the corresponding value of Y by Y", it becomes (q — x') (q — x'') (q — x'"') X Y". .... Now since each of the roots x', x'', x''', &c., is included between p and q, the numerator and denominator of each of the fractions must be affected with contrary signs, and therefore each of these fractions must be negative. But since Y' and Y" must, by art. 282, have the same sign, the fraction is positive. Y' The product of all these fractions is therefore positive, when the number of the fractions is even, that is, when the number of the roots, x', x'', x''', &c., is even; and this product is negative, when the number of these roots is uneven. The values which the given first member obtains by the substitution of Р and q for x must, consequently, be affected with contrary signs in the latter case; and with the same sign in the former case. 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, gives the value of the first member ∞2 + a ∞on-1+ b con-2+ &c. 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, or positive. Again, the substitution of gives, since n is uneven, - ∞2 + a ∞1-1 —b∞1−2+ &c. +m, 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 - co, 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. F 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+b x2 - 2 + &c. ...+m=0, in which n is even. First. To prove that the number of real roots is even. The substitution of 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. = 0 by u and its derivative by U. Find the greatest common divisor of u and U, and, 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. |