Real Roots of Equations.

The real roots of an Equation are so called to distinguish them from imaginary quantities (246), which are sometimes found among the values of x. Real roots may be either positive or negative, rational

or irrational.

(290.) If an Equation whose second member is 0, give results with contrary signs when any two numbers are substituted for the unknown quantity, the Equation will have at least one real root between those ▾mbers.

If in the Equation

x3-3x2+5x-36-0,

we substitute 5 for x, the result will be

125-75+25-36-39:

and if we substitute 2 for x, the result will be

8-12+10-36=―30.

If the value of x vary continually, the value of the first member of the Equation will vary continually. But when a thus varies from 5 to 2, the first member passes through all the values between 39 and —30, and must therefore pass through 0.

There is therefore some number between 5 and 2, which, substituted for x, will make the first member =0, and that number is a root of the Equation.

(291.) Cor. If the two numbers which give results with contrary signs, when substituted for x, differ by unity; the smaller number (regarding both as positive), is the integral part of a root of the Equation.

In the Equation x-12x3-20x2—36=0,

if 13 and 14 be substituted for x, the results will be 1219 and 1532 respectively; hence, 13 is the integral part of a root of the equation.

Again; if -2 and -3 be substituted for x, in, the same equation, the results will be 4 and 189 respectively; hence, -2 is the integral part of another root of the equation.

Irrational and Imaginary Roots.

(292) Irrational and imaginary Roots enter Equations by pairs; so that an Equation will have an even number of such roots, or none

at all.

For, if the Equation be divided by the product of the binomial factors corresponding to all of its roots but two, the quotient put =0, will form a quadratic Equation.

Let a+b be one of the roots of this quadratic; then will a√/b be the other root, since the square root of b is ±√/b, and the two roots, a±√/b, are evidently the two roots of the given equation which were excluded from the preceding product. If therefore an equation contain one irrational root, a+ √/b, it will also contain another, a

-√b.

In a similar manner, it may be shown, that if any odd number of the roots be irrational, there will be also another irrational root;—and the demonstration, it is evident, will be equally applicable to imaginary

roots.

(293.) Cor. An Equation of an even degree, may have all its roots imaginary; but if they are not all so, two of them at least are real.

The precise number of real, and thence of imaginary roots of an Equation, is determined by the following Theorem of STURM-preliminary to which is the definition of

Variations and Permanences of Signs.

(294.) Two consecutive signs in a series of terms, constitute a variation when one of them is + and the other and a permanence when they are both + or both

Thus, in the polynomial

ab+c+x-y-z,

the signs of a and b constitute a variation, of b and c a variation, of c and x a permanence, of x and y a variation, and of y and z a perma

nence

In the given polynomial there are therefore 3 variations and 2 permanences among the signs of the terms.

STURM'S THEOREM.

(295.) This Theorem, which bears the name of its discoverer, ex- · plains a general method of determining the number of real roots, and among them the number of positive and negative roots of any given Equation.

If the Equation has two or more equal roots, these may be found by the method already explained (288); and it is necessary, therefore, to consider only the general case in which the roots are unequal.

The statement of the Theorem involves several particulars.

1. Let the Equation, supposed to have no equal roots,—with all its terms on one side,-be represented by

X=0;

find the first Derivative of X, and denote it by D' (286).

2. Divide X by D', until the remainder is of a lower degree than the divisor, and denote the remainder, with its signs changed, by R'. Divide D' by R', in the same manner, and denote the remainder, with its signs changed, by R"; and so on, as in finding the greatest common measure, until the remainder is independent of x; and denote this last remainder, with its sign changed, by K.

3. Find the row of signs of the values of the Functions

X, D', R', R", .... K

which result from substituting any number p for x, and also from substituting any other number t for x; then

The difference between the number of variations in the first row of signs and the number of variations in the second, will be the number of real roots of the Equation comprised between the numbers p and t.

The whole number of the real roots of the Equation, will thus be determined, if p and t be taken equal to +∞ and co, since all real numbers are included between these limits.

The number of positive roots will be determined by taking p and t equal to + and 0; and the number of negative roots, by taking p and t equal to 0 and

∞.

We will first apply, and then demonstrate this Theorem.

EXAMPLE I.

To determine the number of real roots of the Equation

x3-3x2+5x+7=0,

which is known to have no equal roots.

The derivative of the first member is 3x2-6x+5 (286); and having now the two Functions which correspond to X and D', we must find R, &c., according to the Theorem.

To avoid fractions in the several divisions, we may multiply or divide the Functions by positive numbers, as in finding the greatest common measure, since this will not affect the signs of the Functions; but in this example, the last division will be performed fractionally..

We have then

and shall find

=

Xx3-3x2+5x+7;
D'-3x2-6x +5;

R' -2x-13;
K = −3411.

If we substitute ∞, infinity, for x in these Functions, it is evident that the first term of each polynomial will be greater than the sum of all the other terms; so that the sign of each result will be the same as that of its first term.

The difference between the number of variations in the first row of signs, and the number in the second, is one; the given Equation has then but one real root, and has consequently two imaginary roots (255)

To determine whether the real root is positive or negative.

Demonstration of Sturm's Theorem.

The demonstration of this Theorem depends on several distinct considerations respecting the series of Functions which it involves.

I. Two consecutive Functions in the series X, D', R', R',...K, cannot both vanish, that is, become 0, for the same value of x.

For, denoting by q', q', &c., the quotients of the successive divisions prescribed in the Theorem; then R', R", &c., denoting the successive remainders, with their signs changed, we have

If now D'=0, and R'=0, for the same value of x, Equation (2) would give R"=0; and then Equation (3) would give R""=0; and so on, to the last remainder K=0. But K is independent of x, and cannot therefore reduce to 0 for any value of x.

II. If any one of the Functions between X and K vanish for any value of x, the two adjacent Functions will have contrary signs. Thus, if R'=0, Equation (2) will give D'--R";

that is, the two Functions in the series which are adjacent to the vanishing one, will have equal values but contrary signs.

III. The number of variations of signs of the whole series, X, D', R', R", &c., cannot be affected by the vanishing, or change of signs, of any of the Functions between X and K, for any value of x.

For no two consecutive Functions can vanish at the same time (I); and if any two or more which are not consecutive, could vanish together; each vanishing Function, being interjacent to two others which have contrary signs (II), would form a variation with one, and only one, of them before vanishing, while those two would form a variation with each other afterwards.

Again, since a Function can change its sign only by passing through 0, two consecutive ones cannot change their signs together (I); and when they are not consecutive, their changes of signs will not affect the whole number of variations.

IV. If r be a root of the Equation X=0, the signs of X and D' will form a variation for a value of x which is a little less than r, and a permanence for a value of x which is a little greater than r.

Sincer is supposed x in the Equation X=0, the substitution of r+h for x will result in the same development as the substitution of x+h, in the Function X. This development, we have heretofore seen, is