Imaginary Roots, when Terms are wanting.

supply the place of each deficient term with zero affected

by the same sign as that of the term preceding the deficient All the successions, dependent upon the deficient terms, must in this case be permanences, except the last, which is a variation; so that v + 1 will be the whole number of variations of the given equation, and the number of positive roots cannot, therefore, exceed v + 1.

But if the sign of first, third, fifth, &c. zeros be reversed, all the permanences dependent upon the deficient terms are changed into variations, and the variation is changed into a permanence; so that p+1 will be the whole number of permanences of the given equation; and the number of negative roots cannot, therefore, exceed p + 1.

Hence the whole number of real roots cannot exceed

v + p + 2 = n (m + 1) + 2 = n (m − 1); and, therefore, the remaining (m-1) roots must be imaginary.

204. Theorem. When an even number of successive terms is wanting in an equation, the number of imaginary roots must be at least as great as the number of these deficient terms.

Demonstration. Let m, n, p, v, and s be used as in the preceding article. Let the place of the first deficient term be supplied by zero affected with the same sign as that of the term which follows the deficient terms.

The number of deficient terms is thus reduced to the uneven number m 1; and, as the term preceding the deficient terms is now of the same sign with that of the term following them, the number of imaginary roots of the equation must, by the preceding article, be at least as great as

(m1)+1= m.

Cases in which Cubic Equation has Real Roots.

205. Corollary. Any equation of the third degree may, by art. 170, be reduced to the form

u3 + pu + q = 0

in which the second term is wanting; and this equàtion must, by art. 202, have two imaginary roots, whenever p is positive.

206. Theorem. The equation

u3 — pu + q = 0

must have two imaginary roots, whenever p is positive, and

Demonstration. Since the given equation is of the third degree, it must, by art. 193, have at least one real root. Denote this root by a; and the equation is, by art. 161, divisible by ua; and the quotient is evidently of the form

u2 + au + b ;

so that the other two roots of the given equation are the same with those of the equation

u2 + a ù + b = 0.

Now, by art. 154, the roots of this equation are imaginary, when

and not otherwise.

a2 <4b,

Superior Limit of Positive Roots.

When the given equation has imaginary roots, we have

207. A number, which is greater than the greatest of the positive roots of an equation, is called a superior limit of the positive roots; and one, which is less than the least of the positive roots, is called an inferior limit of the positive roots.

In the same way, a superior limit of the negative roots is a number which, neglecting the signs, is greater than the greatest negative root; and an inferior limit of the negative roots is a number which is less than the least negative root.

208. Problem. To find a superior limit of the positive roots.

Solution. The sum of all the negative terms being equal to the sum of all the positive terms, must exceed each positive term. Let, then, Sbe the greatest negative coefficient of the equation of the nth degree, and m the exponent of the

Superior Limit of Positive Roots.

highest negative term; the sum of the negative terms, neglecting their signs, must evidently be less than that of the series

S+Sx+S x2 + &c. ... + Sxm,

for each term of this series is greater than the corresponding negative term of the equation.

But this series is a geometrical progression of which S is the first term, Srm the last term, and x the ratio; so that its sum is, by example 3, of art.,186,

and must be greater than any positive term, as x", or

x - 1 < x and (≈ — 1 )n — m − 1 < xn — m −1,

we must have

(x − 1)n — m < (x − 1 ) xn — m − 1 < S;

If we, then, denote by L this superior limit of the positive

Limits of Negative Roots.

that is, a superior limit of the positive roots is unity, increased by that root of the greatest negative coefficient, whose index is equal to the number of terms which would precede the first negative term, if no terms were wanting.

209. Problem. To find an inferior limit of the positive roots.

Solution. Substitute in the given equation for 7, the value

x =

1

and find, by the preceding article, a superior limit of the positive values of y, after the equation is reduced to the form of art. 168; and denote this limit by L'.

is an inferior limit of the positive roots of the

given equation.

210. Problem. To find the limits of the negative roots of an equation.

and the positive roots of the equation thus formed are the