448. If the coefficients of an equation be rational quantities, irrational roots of the form a ± √√, or ± √b, enter it by pairs.

This may be proved in the same manner as the preceding proposition, substituting rational and irrational for real and imaginary.

449. The product of a pair of imaginary quantities is always positive. Thus,

and

(± a + b√ − 1) (± a − b √ − 1) = a2 + b2,

(+ b √ − 1) (− b √ − 1) = b2.

450. An equation whose coefficients are real, and whose degree is odd, must always have an odd number of real roots; and it must therefore always have at least one real root, affected with a sign contrary to that of its last term (Art. 441).

451.

If the degree is even, the equation must either have an even number of real roots, or none; but if its last term is negative, it must have at least two real roots, with different signs (Arts. 441, 449).

EXAMPLES.

1. One root of the equation a3 + x2 + 4 = 0 is -2; what are the others?

Ans.

7

2

2. One root of the equation x-12x2+18x+56=0 is 430; what are the others?

3. One root of the

+80 is 222;

is

Ans. 4.

30 and 4. equation x-5x-2x2 + 12 x what are the others?

Ans. 222, -1, and 2.

4. One root of the equation x23 +7 x2 + 20 x + 50 = 0

DESCARTES' RULE.

452. A complete equation cannot have a greater number of positive roots than it has variations of sign; nor a greater number of negative roots than it has permanences of sign.

Let any complete equation have the following signs,

in which there are three permanences and five variations (Art. 433).

If we introduce a new positive root, we multiply by xa, and the signs of the partial and final products may be expressed thus,

However the ambiguous sign is interpreted, there must be at least one variation between 1 and 4, and one between 8 and 10; and since the four variations between 4 and 8 remain as in the original, there are at least six variations in the product. Consequently, the introduction of a positive root has caused the introduction of at least one additional variation; and as this is true of any positive root, there must be at least as many variations of sign as there are positive roots.

In the same way, by introducing the factor xa, it may be shown that there must be at least as many permanences of sign as there are negative roots.

453. If an equation is incomplete, the missing terms should be supplied, with the coefficient ±0 before applying the foregoing rule (Art. 430).

In such an equation, imaginary roots may sometimes be discovered by means of the double sign of 0. Thus, in the equation x+x2+0x+4=0, if we take the upper sign, there is no variation, and no positive root; and if we take the lower sign, there is only one permanence, and but one negative root. Therefore there are two imaginary roots.

In general, whenever the term which precedes the missing one has the same sign as that which follows it, the equation must have imaginary roots. If those terms have different signs, and only a single term is wanting, the equation may, or may not, have imaginary roots, but Descartes' Rule does not detect them. If two or more successive terms of an equation be wanting, the equation must have imaginary roots.

454. In any complete equation, the sum of the number of variations and of permanences is equal to the number of terms less one, or to the degree of the equation (Art. 429). Hence, 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 (Art. 436).

455. A complete equation whose terms are all positive can have no positive root; and one whose terms are alternately positive and negative can have no negative root.

EXAMPLES.

1. What is the limit to the number of positive and of negative roots in the equation x3- 11 x2+43 x — 65 = 0 ? Ans. It can have no negative root, and not more than 3 positive ones.

2. The roots of the equation x30x2 are all real; what signs must they take?

3 x 2 0

Ans. One positive; two negative.

3. The roots of the equation 3 10x+3= 0 are all real; what signs must they take?

Ans. Two positive; one negative.

4. The roots of the equation 3 · 7 x2 + 36 = 0 are all real; what signs must they take?

5. What are the signs of the roots of the equation 4= 0? Ans. One positive; two imaginary.

6. If the roots of the equation — 11x+17x23 + 17 x2 -11x+1= 0 are all real, what signs must they take? Ans. Four positive; one negative.

7. What are the signs of the roots of the equation x2+16x+550?

TRANSFORMATION OF EQUATIONS.

456. The TRANSFORMATION of an equation, as the term is here used (Art. 149), consists in changing the equation into another of the same degree, whose roots shall bear a specified relation to those of the given equation.

457. To transform an equation into another which shall have the same roots with their signs changed

Let the given equation be

-1

2-2

x2 + p x2-1+ q x2-2 +.....+t x2 + u x + v =

= 0,

Put x = -y; then, whatever value x may have, y will have the same, with its sign changed. Substituting this value of x, we have

(—y)"+p(—y)"−1+q(− y)"−2+.....+ty2-uy+v=0,

in which the terms having an odd exponent are negative, and the others positive. If n is even, the equation is in its proper form; but if n is odd, all the signs should be changed; in either case, then, we have

y' —py-1+qyr −2 — . . . . . ± (+ ty2 — uy + v) = 0. Hence, to effect the required transformation, we may simply

Change the signs of the alternate terms.

EXAMPLES.

2, and and 3.

1. The roots of the equation x3 — 7 x + 6 = - 7x+6= -3; find the equation whose roots are

-

0 are 1,

— 1,. — 2,

Here, as the complete equation is x3± 0x2 — 7x+6= 0, we must change the signs of 0x2 and 6; hence the required equation is 3 — 7 x · 60.

2. Two of the roots of the equation - 2 x3 + x - 1320 are 4 and -3; find the corresponding equation whose roots are 3 and —4.

3. If the roots of the equation x3-6x2+3x+10=0 are 5, 2, and -1, what are the roots of the equation 23+6x2+3x100?

458. To transform an equation into another whose roots shall be some multiple of those of the given equation.

y"+pmy11+q m2 yn−2 + .....+tm"-2y2+um"-1y+v m”—0. Hence, to effect the required transformation, we may

Multiply the coefficient of the second term by the given factor, that of the third term by its square, that of the fourth term by its third power, and so on, for the coefficients of the transformed equation.