ber of variations in the product will be greater than in the multiplicand, by what has just been shown. But the product thus obtained is the same as the product of X, and -a; hence, the number of variations in the product of X, and xa is greater than in X.

We have thus shown that when the factor x a is introduced into (2), the resulting equation contains at least one more variation than (2). In like manner it may be shown that when the factor xb is introduced into the resulting equation, at least one more variation is introduced; and so on.

Hence the number of real positive roots of the equation X=0 cannot exceed the number of variations in the signs of its terms. We prove the second part of the theorem as follows:

Suppose (1) to be complete, and let the signs of its alternate terms be changed; then the signs of the roots will be changed (444), the permanences will become variations, and the variations will become permanences. But the number of real positive roots of the resulting equation cannot exceed the number of variations in the signs of its terms; hence the number of real negative roots of the given equation cannot exceed the number of permanences in the signs of its terms.

447. Although the introduction of a positive root will always give an additional variation of signs, it is not true that a variation of signs in the terms of an equation necessarily implies the presence of a real positive root. Thus, the equation,

has 2 variations of signs, and 1 permanence. But its roots are 2 + √ 1, 2 −√1, and 3,

no one being positive and real.

But when the roots are all real, the number of positive roots is equal to the number of variations, and the number of negative roots is equal to the number of permanences.

CARDAN'S RULE FOR CUBIC EQUATIONS.

448. It has been shown (437), that any equation can be transformed into another which shall be deficient of its second term. That is, every cubic equation can be reduced to the form of

and the solution of this equation must involve the general solution of cubics. We make 3p the coefficient of x, and 2q the absolute term, in order to avoid fractions in the following investigations : Assume

(v + y)3

x = v + y ;
+ 3p (v + y) = 2q

then (1) becomes

(2).

Expanding and reducing,

v3 + y3 + 3 (vy + p) (v + y) = 2q. (3).

Now as the division of x into two parts is entirely arbitrary, we are permitted to assume that

If we obtain the value of y from (4), and substitute it in (5), we shall have, after reducing,

But by hypothesis xv+y; hence, taking the sum of the cube roots of (7) and (8),

x = (q + √ q2 + p3)3 + (q − √ q2 + p3)‡ (4),

which is Cardan's formula for cubic equations.

449. When p is negative, in the given equation, and its cube numerically greater than q2, the expression √q2 + p3 becomes imaginary; this is called the Irreducible Case. We must not conclude, however, that in this case the roots of the equation are imaginary; for, admitting the expression √q2 + p3 to be imaginary, it can be represented by a √-1; whence the value of in formula (4) becomes

= (q + a √ − 1)$ + (q−a√=1)§ . . (1);

x=

or, z=q3 (1 + " √− 1)* + q3 (1 — ¦ √=1) *

x

(2);

(3).

Now by actually expanding the two parts in the second member of (3), and adding the results, the terms containing √-1 will disappear and the final result will be real. In the irreducible case all the roots of the equation are real; formula (A) is therefore practically applicable only when two of the roots are imaginary. In this case the real root can be found directly by the formula; the equation may then be depressed, by division, to a quadratic, which will give the two imaginary roots.

To transform this equation into another deficient of its 2d term, according to (437), put xy+; and we shall have for the transformed equation,

=

y3 — Zy = 344.

To apply the formula to this equation, we have

3p, or p};

=

子;

√ q2 + p3 = √(177)2 — (7)3 = ± 14.

y = (178 ± 171)3 + (3} = ¿Y)$ = } + ƒ = }.

F 17)

x=+=5, the real root.

Dividing the given equation by x-5, we obtain for the depressed equation

whence,

x2 - 2x + 4 = 0;

Hence the three roots are 5, 1+-3, and 18.

2. Given 3+6x=88, to find the values of x.

To apply the formula, we have

or,

√1936 + 8 = ± 44.090815+.

x = (44 + 44.090815)* + (44 — 44.090815)*;

The depressed equation will be x2+4x-22 = 0; whence,

x=2±31-2;
3√2;

and the three roots are 4, 2 + 3 √2, and - 2 — 3 √ — 2.

5.6, to find one value of x.

This example presents the irreducible case; the solution, by the method of series, is as follows:

We have

or,

or,

p=2, q= 2.8; hence,

x = -
(2.8 + √7.84 — 8) + (2.8 — √√/7.84 — 8)$;
x = (2.8 + .4 √− 1)3§ + (2.8 — .4 √ — 1)3 ;

Put

=

b=√1; then 62, 64 1 × &•

=

Also, (1 +‡ √− 1)3 = (1 + 6)3 ; (1 − ‡ √− 1)$ = (1 — b)§.

By the binomal theorem

2.004569 ; x = (2.004569) †2.8 = 2.82535, Ans.

4. Given x3 - 6x - 6 = 0, to find one value of x.

Ans. x= √2 + 1/4 = 2.8473+.

5. Given 3 + 9x — 60, to find one value of x.

6. Given x+6x2

/9+3.63783+.

13x + 24 0, to find the values of x.

Ans. x 8, 1+V2, or 1-√2.

NUMERICAL EQUATIONS OF HIGHER DEGREES. 405

SECTION IX.

SOLUTION OF NUMERICAL EQUATIONS OF HIGHER

DEGREES.

LIMITS OF REAL ROOTS.

∞. But in

450. All positive roots of an equation are comprised between O and∞, and all negative roots between 0 and the solution of numerical equations of higher degrees, it is necessary to be able at once to assign much narrower limits. As preliminary to this, we will first show how an equation is affected by substituting for the unknown quantity numbers greater or less than the roots, and numbers between which the roots are comprised.

451. If an equation, in its general form, be regarded as the product of the binomial factors formed by annexing the roots, with their opposite signs, to x, we observe that the sign of this product cannot be affected by the imaginary roots. For, according to 445, if an equation have one root in the form of a + √ — b, it will have another in the form of a b. But we have

-

(x — a — √—b) (x − a + √ − b ) = (x — a)2 + b,

a result which is in all cases positive.

452. Let a, b, c, d, etc., be the real roots of an equation, arranged in the order of their algebraic values; then the equation may be represented as follows:

(x − a) (x — b) (x — c) (x — d) . . . . = 0.

If we substitute h for x, the first member will become

Now if h be less than the least root, a, every factor will be negative; and the whole product will be positive or negative, according as the number of factors is even or odd. But the number of factors is equal to the degree of the equation (427); hence,

1. If a number less than the least root be substituted for x in an

A*