Find the fifth roots of +1 and the sixth roots

If w be one of the complex cube roots of + 1,

Show that I

w2 is one of the twelfth roots of +1.

(8). The twentieth roots of +1 are the successive powers, from the first to the twentieth inclusive, of

I.

and that σ is therefore a tenth root of
even powers of σ are the tenth roots of +1.

XIX.

= 0,

Show that the

POLYNOMIALS AND EQUATIONS.

115. Definition of Polynomial. An algebraic expression of the form

in which a, a,, . . . an are any quantities not involving z, and in which the exponents of z are all integers, is called a rational, integral polynomial in z. In what follows it will be sufficient to speak of it more briefly as a polynomial, the qualifying adjectives being understood. The highest exponent of a contained in it is its degree.

116. Roots of Equations. The investigation of Art. 113 solves the problem of finding what are called the roots of the equation

zn = ω,

in which w is a known complex quantity, and shows that such an equation, which would commonly present itself in the binominal form

has exactly n roots.

azn+b=0,

[b/a = -w]

If additional terms containing powers of z lower than the nth be introduced into this equation, the problem of its solution becomes at once difficult, or impossible. In fact, the so-called algebraic solution of a general algebraic equation of a degree higher than the fourth, that is, a solution involving only radicals and having a finite number of terms, is known to be impossible.*

A discussion of the methods that may be employed in solving equations is beyond the intended scope of the present work, but the so-called fundamental theorem of algebra (Art. 120), accompanied by those propositions that are prerequisite to its demonstration, find a fitting place here.

117. The Remainder Theorem. If a polynomial of the nth degree in z be divided by zy, the remainder, after n successive divisions, is the result of substituting a for y z in the polynomial.

Let f (2) denote the polynomial,

f(z) = a + a ̧ z + a ̧22+

2

+an zn,

α

* Proved to be so by Abel: Journal für die reine und angewandte Mathematik (1826), Bd. I, pp. 65-84, and Œuvres completes de N. H. Abel, nouv. ed., Vol. I, pp. 66-94.

and let the division by z-y be performed. It is obvious. that the remainder after the first division will be of a degree lower by than the dividend, that each succeeding remainder will be of a degree lower by 1 than its predecessor, and that therefore the nth remainder will not involve z. If the final quotient be denoted by Q and the final remainder by R, then

f(z)

ƒ (2) = 0 +

2

R

2 γ

in which R does not involve z; whence, by multiplying by 2-Y,

f(z) = Q(≈ − y) + R.

This equation has the properties of an identity,* and in it may therefore have any value whatever. Accordingly, let y be substituted for z, and let the result of this substitution in the polynomial be denoted by ƒ (y); then

in which R remains unchanged from its former value, and Q, being now a polynomial in y, is finite.

Hence

Q. E. D.

If the remainder obtained in dividing f(z) by z— γ vanish, then f(y)=0 and y is said to be a root of the equation ƒ(z)=0.

Hence:

*This may be shown by actually evolving it in specific instances. Thus, if the process here described be applied to the quadratic a+a1z+az2, the result is

а ̧+a ̧≈+a ̧22==(α1+a ̧y+a2z) (z−y)+а2+a1y+a2y2,

and the principle of this procedure is obviously general, and independent of the degree of the polynomial. It should be observed that the identity does not depend upon the process of division; we might, in fact, produce it by the processes of addition, subtraction and rearrangement of terms. The division process is used as a convenience, not by necessity.

(i). If f(z) be exactly divisible by zy, y is a root of the equation f(z) = 0.

Conversely, the remainder will vanish if f(y)=0.

Hence:

(ii). Ify be a root of the equation f(z) = 0, then ƒ (z) is exactly divisible by z — γ.

118. Argand's Theorem. If for a given value of z the polynomial of the nth degree,

have a value w different from zero, its coefficients a, a,, a,... an being given quantities, real, imaginary, or complex, there exists a second value of z, of the form x + iy, for which the polynomial has another value w such that

tsr wtsr w。.

The following demonstration of this theorem is a modification of Argand's original proof.*

Let zo be the given value of ≈ and let wo be the resulting 20 value of the polynomial, different from zero, so that

Add to 。 an arbitrary complex increment z, whose tensor and amplitude are disposable at pleasure, and let w be the resulting value of the polynomial, so that

w=a。 + a1(。 + z) + α2 (。 + 2)2 + . . . + an (3。 + z)”.

W=

2

If the several powers of the binominal z + %, in this equation be expanded by actual multiplication, or by the

* Argand: Essai sur une manière de représenter les quantités imaginaires dans les constructions géométriques (Paris, 1806), Art. 31. The demonstration was reproduced by Cauchy, in the Journal de l'École Royale Polytechnique (1820), Vol. XI, pp. 411-416, in his Analyse algébrique (1821), ch. X, and again in his Exercises d'analyse et de physique mathématique, Vol. IV, pp. 167-170.

binominal theorem, and its terms be then arranged according to the ascending powers of z, it may be written in the form

%0

in which b1, b, ... bn, involve z but not z. By hypothesis an is not zero, but any or all of the coefficients b1, b2 ... b, may possibly vanish. Let bm be the first that does not vanish, b, standing for the same thing as an, so that w=w+bm zm+ (bm+1+bm+2% +.... +b12 zn−m−1) zm+1, and let

and

bm=a cis a,

zn−m−1

bm + 1 + b m + 23 + ... + b12 = "-m-1b cis ß,

z=r cis 0,

the quantities a, a, b, 3, r and being real. Then

w=w+arm cis (a + m0) + brm+1 cis { ß+ (m + 1) 0}.

1

Since the length of any side of a closed polygon cannot be greater than the sum of the lengths of all the other sides, or in other words, since the tensor of a sum cannot be greater that the sum of the tensors of the several terms (Art. 58),

...

b≤tsrbm+1+r tsr bm+2 + . . . + rn-m-1 tsr b„.

...

Hence, by diminishing sufficiently b may be made to differ from tsr bm+1 by an arbitrarily small quantity, and a maximum limit to the variation of r may be assigned such. that b shall not exceed a fixed finite value. It follows that ✓ may be taken so small that

bra, or hym+1<arm.