Chapter 7
The Fundamental Theorem of Algebra

52. The General Theorem. If

w = a0zn + a 1zn1 + a 2zn2 + + a n1z + an,

where n is a positive integer, and a0,a1,,an any numbers, real or complex, independent of z, to each value of z corresponds a single value of w.

We proceed to demonstrate that conversely to each value of w corresponds a set of n values of z, i. e. that there are n numbers which, substituted for z in the polynomial a0zn + a 1zn1 + + a n, will give this polynomial any value, w0, which may be assigned.

It will be sufficient to prove that there are n values of z which render a0zn + a 1zn1 + + a n equal to 0, inasmuch as from this it would immediately follow that the polynomial takes any other value, w0, for n values of z; viz., for the values which render the polynomial of the same degree, a0zn + a 1zn1 + + (a n w0), equal to 0.

53. Root of an Equation. A value of z for which a0zn + a 1zn1 + + a n is 0 is called a root of this polynomial, or more commonly a root of the algebraic equation

a0zn + a 1zn1 + + a n = 0.

54. Theorem. Every algebraic equation has a root.

Given w = a0zn + a1zn1 + + an.

Let |w| denote the modulus of w. We shall assume, though this can be proved, that among the values of |w| corresponding to all possible values of z there is a least value, and that this least value corresponds to a finite value of z.


pict
Fig. 4.


Let |w0| denote this least value of |w|, and z0 the value of z to which it corresponds. Then |w0| = 0.

For if not, w0 will be represented in the plane of complex numbers by some point P distinct from the null-point O.

Through P draw a circle having its centre in the null-point O. Then, by the hypothesis made, no value can be given z which will bring the corresponding w-point within this circle.

But the w-point can be brought within this circle.

For, z0 and w0 being the values of z and w which correspond to P, change z by adding to z0 a small increment δ, and let Δ represent the consequent change in w. Then Δ is defined by the equation

(w0 + Δ) = a0(z0 + δ)n + a1(z0 + δ)n1 + a2(z0 + δ)n2 + + an1(z0 + δ) + an.

On applying the binominal theorem and arranging the terms with reference to powers of δ, the right member of this equation becomes

a0z0n + a1z0n1 + + an1z0 + an + [na0z0n1 + (n 1)a1z0n2 + + an1]δ +   terms involving δ2δ3, etc.

   But w0 = a0z0n + a 1z0n1 + + a n1z0 + an. Δ = [na0z0n1 + (n 1)a 1z0n2 + + a n1]δ +   terms involving δ2δ3, etc.

Let ρ(cos𝜃 + isin𝜃) be the complex number

na0z0n1 + (n 1)a 1z0n2 + + a n1,
expressed in terms of its modulus and angle, and
ρ(cos𝜃 + isin𝜃)
the corresponding expression for δ. Then

Δ = ρ(cos𝜃 + isin𝜃) × ρ(cos𝜃 + isin𝜃) +   terms involving ρ2ρ3, etc. = ρρ[cos(𝜃 + 𝜃) + isin(𝜃 + 𝜃)] +   terms involving ρ2ρ3, etc.       § 49.

The point which represents ρρ[cos(𝜃 + 𝜃) + isin(𝜃 + 𝜃)] for any particular value of ρ can be made to describe a circle of radius ρρ about the null-point by causing 𝜃 to increase continuously from 0 to 4 right angles.

In the same circumstances the point representing

w0 + ρρ[cos(𝜃 + 𝜃) + isin(𝜃 + 𝜃)]
will describe an equal circle about the point P and, therefore, come within the circle OP.

But by taking ρ small enough, Δ may be made to differ as little as we please from ρρ[cos(𝜃 + 𝜃) + isin(𝜃 + 𝜃)],24and, therefore, the curve traced out by P (which represents w0 + Δ, as 𝜃 runs through its cycle of values), to differ as little as we please from the circle of centre P and radius ρρ.

Therefore by assigning proper values to ρ and 𝜃, the w-point (P) may be brought within the circle OP.

The w-point nearest the null-point must therefore be the null-point itself.25

55. Theorem. If α be a root of a0zn + a1zn1 + + an, this polynomial is divisible by z a.

For divide a0zn + a1zn1 + + an by z a, continuing the division until z disappears from the remainder, and call this remainder R, the quotient Q, and, for convenience, the polynomial f(z).

Then we have immediately

f(z) = (z α)Q + R,
holding for all values of z.

Let z take the value α; then f(z) vanishes, as also the product (z α)Q.

Therefore when z = α, R = 0, and being independent of z it is hence always 0.

56. The Fundamental Theorem. The number of the roots of the polynomial a0zn + a1zn1 + + an is n.

For, by § 54, it has at least one root; call this α; then, by § 55, it is divisible by z α, the degree of the quotient being n 1.

Therefore we have

a0zn + a 1zn1 + + a n = (z α)(a0zn1 + b 1zn2 + + b n1).

Again, by § 54, the polynomial a0zn1 + b1zn2 + + bn1 has a root; call this β, and dividing as before, we have

a0zn + a 1zn1 + + a n = (z α)(z β)(α1zn2 + c 1zn3 + + c n2).

Since the degree of the quotient is lowered by 1 by each repetition of this process, n 1 repetitions reduce it to the first degree, or we have

a0zn + a 1zn1 + + a n = a0(z α)(z β)(z γ)(z ν),

a product of n factors, each of the first degree.

Now a product vanishes when one of its factors vanishes (§ 36, 3, Cor.), and the factor z α vanishes when z = α, z β when z = β,,z ν when z = ν. Therefore a0zn + a0zn1 + + an vanishes for the n values, α,β,γ,ν, of z.

Furthermore, a product cannot vanish unless one of its factors vanishes (§ 36, 4, Cor.), and not one of the factors z α,z β,,z ν, vanishes unless z equals one of the numbers α,β,ν.

The polynomial has therefore n and but n roots.

The theorem that the number of roots of an algebraic equation is the same as its degree is called the fundamental theorem of algebra.