Number of Real Roots.

CHAPTER VIII.

Resolution of Numerical Equations.

SECTION I.

Number of Real Roots of Equations.

190. Theorem. When an equation is reduced, as in art. 160, and the values of its first member, obtained by the substitution of two different numbers for its unknown quantity, are affected by contrary signs, the given equation must have a real root comprehended between these two numbers.

Demonstration. For, if the value of the less of the two numbers, which are substituted for the unknown quantity is supposed to be increased by imperceptible degrees until it attains the value of the greater number, the value of the first member must likewise change by imperceptible degrees, and must pass through all the intermediate values between its two extreme values. But the extreme values are affected with opposite signs, so that zero must be contained between them, and must be one of the values attained by the first member; that is, there must be a number which, substituted in the first member, reduces it to zero, and this number is consequently a root of the given equation.

191. Corollary. If the given equation has no real root, the value of its first member will always be af

1

Number of Real Roots between two given Numbers.

fected by the same sign, whatever numbers be substituted for its unknown quantity.

192. Theorem. When an uneven number of the real roots of an equation are comprehended between two numbers, the values of its first member obtained, by substituting these numbers for x, must be affected with contrary signs; but if an even number of roots is contained between them, the values obtained from this substitution must be affected with the same sign.

Demonstration. Denote by a, b, c, &c. all the roots of the given equation which are contained between the given numbers pand q; the first member of the given equation must, by art. 162, be divisible by

(x - a) (x - b) (x - c) &c.

If we denote the quotient of this division by Y, the equation

Y=0

gives all the remaining roots of the given equation, so that Y = 0

cannot have any real root contained between p and q.

The given first member being, therefore, represented by (x - a) (x - b) (x - c).... X Y

becomes

(p-a) (p-b) (p - c) .... X Y',

when we substitute p for x, and denote the corresponding value of Y by Y'; and when we substitute q for x, and denote the corresponding value of Y by Y", it becomes

(q-a) (q-b) (q-c).... × Y".

The quotient of these two results is

Now since each of the roots a, b, c, &c. is included between p and q, the numerator and denominator of each of

the fractions

q-a

p-ap-bp-c

9-69&c.

c'

must be affected with contrary signs, and therefore each of

these fractions must be negative.

But since Y' and Y" must, by art. 190, have the same sign, the fraction

is positive.

Y
Y"

The product of all these fractions is therefore positive, when the number of the fractions

is even, that is, when the number of the roots a, b, c, &c. is even; and this product is negative, when the number of these roots is uneven. The values which the given first member obtains by the substitution of p and q for a must, consequently, be affected with contrary signs in the latter case; and with the same sign in the former case.

193. Theorem. Every equation of an uneven degree, has at least one real root affected with a sign contrary to that of its last term, and the number of all its

Number of Real Roots of an Equation of an Odd Degree.

Demonstration. Let the equation be

χη + α χη−1 + &c.... + m = 0,

in which n is uneven.

First, to prove that there is a real root, and that the number of real roots is uneven. Every real root must be contained between + ∞ and 00. Now the substitution of

x = 0,

gives the value of the first member

: + an-1 + bon-2 + &c.... + m;

the first term of which is infinitely greater than any other term, or than the sum of all the other terms. The sign of this result is therefore the same as that of its first term, or positive.

+ m,

∞" + an-1-bon-2 + &c. which may be shown by the above reasoning to be negative.

The given equation must then, by art. 191, have at least one real root, and by art. 192, the number of its real roots must be uneven.

Secondly. To prove that one, at least, of the real roots is affected with a contrary sign to that of the last term. The substitution of

reduces the given first member to its last term m.

Comparing this with the above results, we see that, if m is positive, the given equation must, by art. 190, have a real root contained betwen 0 and ∞, that is, a negative root; but if m is negative, there must be a real root contained between 0 and + ∞, that is, a positive root; so that there must always be a root affected with a sign contrary to that of m.

Number of Real Roots of Equations.

194. Theorem. The number of real roots, if there are any, of an equation of an even degree must be even, and if the last term is negative, there must be at least two real roots, one positive and the other negative.

Demonstration. Let the equation be

xn + axn-1 + b xn-2 + &c....+m 0,

in which n is even.

First. To prove that the number of real roots is even. The substitution of

gives for the value of the first member

∞* + an-1 + b2-2 + &c. ... + m,

gives for the value of the first member

cona con-1+ 6-2 + &c. ... + m,

which is also positive.

Hence, if the given equation has any real root there must, by art. 192, be an even number of them.

Secondly. To prove that when m is negative, there must be two real roots, the one positive, the other negative. The substitution of

reduces the given first member to its last term m, and this result is therefore negative in the present case.

Comparing this with the above results, we see that there must be a root between 0 and + ∞, and also one between 0 and ∞; that is, the given equation has two roots, the one positive and the other negative.