and let the division by zy 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

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 f(y); then

ƒ(y)= Q(r− y) + R,

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

Hence

Q. E. D.

γ

If the remainder obtained in dividing f(z) by 2-y vanish, then f(y) =o and y is said to be a root of the equation f(z) = o. Hence:

0.

*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 a ̧±à ̧ï+a ̧22 — (a2+a2y+a2z) ( z − y) +α2+α1y+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 f(z) is exactly divisible by zy.

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 z and let wo be the resulting value of the polynomial, different from zero, so that

Add to z。 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) + a2 (。 + 2)2 + . . . + an (3。 + z)”.

W=

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

w=a+a ̧%+a2z2+a3z3+ +an zn

...

+bn-1zn−1 + anzn,

in which b1, b,... bn-, involve z an is not zero, but any or all b- may possibly vanish.

but not z. bn, involve z

...

but not z. By hypothesis of the coefficients b1, b2, Let bm be the first that does

not vanish, b1 standing for the same thing as an, so that

n

w=w+bm zm +(6m+1+bm+2% +... +b1 zn−m−1) zm+1,

the quantities a, a, b, 3, r and 0 being real. Then

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

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 bn.

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 be taken so small that

r may

bra, or hym+1 <arm.

Let P., P' and P (Fig. 45) be the respective affixes of w ̧, wo+arm cis (a+m0) and w, and let and r, which are at our disposal, be so chosen that

and if this disposition of r be not sufficient to make br.m+1 <arm, let r be still further diminished until

I

br <a

Then, since cis (amp w)= vsr w。 (Art. 60),

w= w ̧— arm cis (amp w ̧) + brm+1 cis { ß + (m + 1) 0 } arm) vsr w。 + brm+1 cis { ß + (m + 1) 0}, brm+1 < arm < tsr w。.

and

(tsr w

In accordance with these relative determinations of tensors and amplitudes the positions of Po, P' and P in the w-plane (Fig. 45) are as follows:

Po, the affix of w。, lies upon a circumference whose center is O and radius OP。.

P

Fig. 45.

X

P', the affix of (tsr warm) vsr wo, lies upon OP between 0 and P.

P, the affix of w, lies upon a circumference whose centre is P' and whose radius is less than P'P。.

This latter circumference is therefore wholly within that upon which P. lies, and P is nearer to O than is P。. But

OP= tsr w。 and OP= tsr w;

O

hence

tsr w

tsr wo.

Q. E. D.

When all the coefficients of z in the expansion of w except an vanish (bn=an), the point P' is coincident with P. The final result in the foregoing demonstration

remains.

119. Every Algebraic Equation has a Root. There exists at least one value of z, real, imaginary, or complex, for which the polynomial of the nth degree,

vanishes, the coefficients a, a,, ... a, being given quantities, real, imaginary or complex.

This theorem is commonly stated in the briefer form: Every algebraic equation has a root. It is an immediate consequence of Argand's theorem.* For, if it be assumed

that there is no value of z for which the polynomial vanishes, and if w。 be that value of the latter whose tensor is the least possible, this hypothesis is at once contradicted by Argand's theorem which asserts that there is another value of z and a corresponding value of w such that

tsr wtsr w。•

Hence, tsr w must have zero as its least value, and for such a value the polynomial vanishes.

I20. The Fundamental Theorem of Algebra. A polynomial of the nth degree, such as

a+a1z +α2≈2 + + anz",

whose coefficients a, a, ... an are given real, imaginary, or complex quantities, is equal to the product of n linear

*The two propositions were not segregated by Argand; both were proved by him in the same paragraph (loc. cit Art. 31).