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

Let f(z) stand for the polynomial. By Art. 119 f(z) = o has at least one root; let that root be Y1. Then, by Art. 117, f(z) is divisible by zy, and may be expressed in the form

f(z)=(z−y)f("),

in which f, (2) is a polynomial of the degree n — 1. I. The equation f,(z) =o has also a root; let this root be y Then, as before,

ƒ1(2) = (≈ — Y2) ƒ1⁄2 (Z),

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

f(z)=(z−y)(z−y)f, (z).

This process repeated n times will produce a quotient of the degree n — no, that is, a quantity independent of z. Hence, ƒ(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 (≈ — 7.) (≈ — Y1⁄2).... (≈ — Y»).

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 ƒ(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 z. Thus f(z) will vanish for the n values YI, Y2) .., 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+y) be a polynomial in xiy having all its coefficients real, and if

then

ƒ (x + iy)=X+iY,

f(xiy)=X-iY.

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

then

f(a + ib) = 0,

f(a―ib) = 0.

(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:

P"

Let the straight line SP, while remaining transversal to the two intersecting straight lines OP, OS, move parallel to a fixed direction with a speed proportional to its perpendicular distance from O. Since OP and OS are proportional to this distance, P and S move also with speeds respectively proportional to to their distances from O.

Let

P

Q"

move along a straight line through O with a constant speed μ, and let the following sets of values correspond respectively, as designated by the accents:

OP, OP, OP" = x, x', x",

Finally, let OS' and OJ be each equal to the linear unit and let

speed of P at J=λ.

Ift represent the time it takes SP to pass to the position S"P", then designating ratio of distance traversed to timeinterval as average speed (abbreviated to av. sp.), we have av. sp. of S= (s'' — s) / t, av. sp. of P= (x" — x)|t,

and

(av. sp. of S) / (av. sp. of P)= (s′′ — s) / (x′′ — x).

This proportion remains, however small the interval be, and the limit of average speed, as the interval approaches zero, is the actual speed at the beginning of the interval. It follows that

and speed of S at unit's distance from O is λ. But by the definition of an exponential (Art. 23), since the positions Q', Q" correspond respectively both to P', P'' and to S', S",

... expmy'' / expmy′ = expm (y” — y′).

END OF THE UNIPLANAR ALGEBRA.