Number of Imaginary Roots; of Real Positive Roots.

195. 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.

196. 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.

Demonstration. The substitution of

X

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

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. 191, be even; and if this last term is negative, the number of these roots must be uneven.

197. Theorem. An equation cannot have a greater number of positive roots than there are VARIATIONS in the signs of its terms, nor a greater number of negative roots than there are PERMANENCES of these signs.

Demonstration. The truth of this proposition would be demonstrated, if it could be shown that the multiplication of the first member of an equation by a factor x - a, corresponding to a positive root, must introduce at least one

Number of Positive and Negative Roots.

variation, and that the multiplication by a factor x + a, must introduce at least one permanence.

Let the general equation of the nth degree be

x + Axn−1 + B xn−2 + .... + L x + M = 0.

in which the signs succeed each other in any manner what

ever.

If we multiply this equation by x - a, it becomes xn+1+Ax± Bx-1± Cx-2 ±±Mx

FLM=0.

The signs in the upper line of this product are the same as in the given equation, while those of the lower line are the reverse of those of the given equation advanced one rank towards the right.

If, then, we proceed from the first to the last term of the product, we must find the same changes of signs as in the given equation, so long as we can remain in either of these lines.

But we are forced to descend from the upper line to the lower line, as soon as we come to a term in which the lower number is larger than the upper one, and has the opposite sign. In this case, a new variation is clearly introduced, for the lower sign is, as before remarked, the reverse of the preceding upper sign; and, all the remaining signs of the lower line being the reverse of the preceding ones of the upper line, we must find the same changes of signs as we should have found in the upper line.

If, however, we should, after this, come to a term in which the upper number is the greater, and has the reverse sign of the lower one, we must reascend to the upper line. In this case, the order of the signs must evidently be the same as it would be, if, in the lower line this term were omitted, and the following signs reversed. But with the omission of a

Number of Positive and Negative Roots.

term, two successions of signs must be lost, one of which at least is here a permanence; for the lower sign of the omitted term, being the reverse of the upper sign, must be the same as its succeeding sign in the lower line. Not more than one variation can, therefore, be lost in ascending to the upper line, and this is replaced by the variation which is introduced in descending again to the lower line; also since the last sign is in the lower line, we must descend again to the lower line.

Each factor, corresponding to a positive root, must then introduce a new variation, so that there must be as many variations as there are positive roots.

In the same way, it may be shown that each factor, as x + a, corresponding to the negative root - a, must introduce at least one new permanence, so that there must be as many permanences as there are negative roots.

198. Corollary. The whole number of succession 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.

199. Scholium. Whenever a term is wanting in an equation, its place may be supplied by zero, and either sign may be prefixed.

200. Corollary. When the substitution of + 0 for a term which is wanting gives a different number of

Zero substituted for a Term which is wanting.

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.

201. 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.

Demonstration. 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 - 0 gives two variations. The reverse is the case when both these terms are negative. The equation must therefore, in either case, have imaginary roots.

202. Theorem. When two or more successive terms of an equation are wanting, the equation must have imaginary roots.

Demonstration. For the second deficient term may be supplied with zero affected by the same sign as that of the term preceding the deficient terms; and the first deficient term is then preceded and followed by terms having the same sign, so that there must, by the preceding article, be imaginary roots.

203. Theorem. When an uneven number (m) of successive terms is wanting in an equation, the number of imaginary roots must be at least as great as (m+1), if the term preceding the deficient terms has the same sign with the term following them; and the number of imaginary roots must be at least as great as (m - 1), if the term preceding the deficient

Imaginary Roots, when Terms are wanting.

terms has the reverse sign of the term following them.

Demonstration. Let n denote the degree of the equation, p the number of permanences of the given equation which are independent of the deficient terms here considered, and v the number of variations; the number of successions which are dependent upon these deficient terms must be (m+1); so that, as in art. 198,

or

v+p+m+ 1 = n,

v + p = n- (m + 1).

First. If the sign of the term preceding the deficient terms is the same with the sign of the term following them; supply the place of each deficient term with zero affected by this same sign. All the successions, dependent upon the deficient terms, must in this case be permanences; so that v will be the whole number of variations of the given equation, and the number of positive roots cannot therefore exceed v.

But if the sign of every other zero beginning with the first is reversed, namely, of the first, third, fifth, &c., all the permanences dependent upon the deficient terms are changed into variations; so that p will be the whole number of permanences of the given equation, and the number of negative roots cannot therefore exceed p.

Hence the whole number of real roots cannot exceed v+p=n-(m + 1);

and, therefore, the remaining (m + 1) roots must be imaginary.

Secondly. If the sign of the term preceding the deficient terms is the reverse of the sign of the term following them;