Let Po, 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

a + mo

=

amp w+π,

and

arm <OP ̧;

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

br <a

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

w= w ̧— arm cis (amp w ̧) + brm+1 cis { ß + (m + 1) 0}

and

=

(tsr w ̧ — arm) vsr w。 + bṛm+1 cis { ß + (m + 1) 0}, brm+1 < arm < 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:

P

Fig. 45.

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

When all the coefficients of z in the expansion of w except a, vanish (ban), 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 wo 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.

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

a+a1z+a2 =2 + + anz",

I

2

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).

factors multiplied by the coefficient of the highest power of z in the polynomial.*

Let f() stand for the polynomial. By Art. 119 f(z) = o has at least one root; let that root be γι Then, by Art. 117, f(z) is divisible by z―y, and may be expressed in the form

f()=(z−y)f("),

in which f,(z) is a polynomial of the degree n — I. The equation ƒ, (≈)=o has also a root; let this root be y2. Then, as before,

f(z)=(z−y)f, (2),

in which f,(z) is a polynomial of the degree n — 2; whence

f(≈)=(z−y)(z−y)f, (~).

This process repeated n times will produce a quotient of the degree n―n=o, that is, a quantity independent of z. Hence, f(z) may be expressed as the product of ʼn linear factors and a factor independent of z; and because the coefficients of the highest powers of z on the two sides of our equation must be identical, this last factor is an

f(z) = an (z — y1) (≈ — y1⁄2) . . . . (≈ — Yπ).

....

Thus

Q. E. D.

*The first demonstration of this celebrated theorem was given by Gauss in his Doctor-Dissertation which bore the title Demonstratio nova theorematis omnem functionem algebraicum rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse, and was published at Helmstädt in 1799. He presented a second proof in December, 1815, and a third in January, 1816, to the Königliche Gesellschaft der Wissenschaften zu Göttingen. (See Gauss: Werke, Bd. III.) For further notices concerning this theorem see the following: H. Hankel: Complexe Zahlen, pp. 87-98; Burnside and Panton: Theory of Equations, 2d ed., pp. 442-444; Baltzer: Elemente der Mathematik, 6. Aufl., Bd. I, p. 299.

As a corollary of this theorem we have: An equation of the nth degree has n and only n roots. For, the condition necessary and sufficient in order that f(z) may vanish, is that one of its linear factors shall be zero, and the putting of any one of its linear factors equal to zero gives one and only one value of 2. Thus f(z) will vanish for the n values , Yn of z, and for no others.

121. Agenda. Prove the following theorems concerning polynomials:

(1). Every polynomial in x + iy can be reduced to the form X+iY.

(2). If f(x+iy) be a polynomial in x + ¿y having all its coefficients real, and if

then

f(x+y)=X+iY,

f(x — iy) = X— iY.

(3). If all the coefficients of the polynomial ƒ(z) be real, and if

(4). In an algebraic equation having real coefficients imaginary roots occur in pairs.

APPENDIX.

SOME AMPLIFICATIONS.

Art. 23, page 40.

The notation by does not here presuppose any knowledge of the fact, easily proved as indicated in Art. 38, that by has the usual arithmetical meaning when b and y are numbers.

Art. 24, page 42.

The following systematic arrangement of the steps of the proof of the first proposition of page 42 will aid the student to a clearer apprehension of it.

The following alternative proof of the law of indices, though longer than that given on pages 43-44, has the merit of greater explicitness: