Let F be a field with S the set of its elements and its two operations denoted by + and juxtaposition for addition and multiplication. Both adition and multilication are associative. Addition is commutative but multilication is not necessarily so. A polynomial over F is of the form

P(C, k) = c_nkⁿ + c_n-1k^n-1 + … + c₁k + c₀

where the coefficients c_j and the base of the polynomical k belong to S and n is a nonnegative integer. The largest power n is called the degree and the smallest power m with a nonzero coefficient will be called the minimum power of the polynomial, The difference (n−m) will be called the degree span of the polynomial.

The field has additive and multiplicative identities which will be denoted as z and e, respectively . There are times in which 0 and1 will be used for these identities.

The multiplicative identity has an additive inverse which will be denoted as −e. Thus there is an element (k+(−e)).

Theorem 1: The polynomials (kⁿ+(−e)) are equal to (k^n-1 + k^n-2 + … + k + e)(k+(-e)) for all positive n.

(kⁿ⁻¹ + kⁿ⁻² + … + e)(k+(−e))
                 = kⁿ + kⁿ⁻¹ + kⁿ⁻¹ + … + k)                        
+(−e) (kⁿ⁻¹ + kⁿ⁻² + … + k + e)
  _____________________________________________________
  = kⁿ + (− e)

This makes use of the fact that x+(-e)x is equal to the additive identity z=0. This follows from x+(-e)x= ex+(-e)x=(e+(-e))x=0x=0=z.

Thus, in particular, (k+(-e)) is an exact factor of (kⁿ + (− e)) for all positive integer values of n.

Let σ(P(C, k)) denote the sum the coefficients of a polynomial. Then

Theorem 1: The polynomial (k+(-e)) is an exact factor of [P(C,k)+(−σ(P(C, k))].

Proof:

[P(C,k) + (−σ(P(C, k))] = Σ c_jk^j + (− Σ c_j)
= Σ c_j(k^j+(−e))
(k+(-e)) exactly divides
each (k^j+(−e))
therefore it exactly divides the sum

Note that σ(P(C, k)) = P(C, e).

Sums of Polynomials

Let N and M be two polynomials to the same base k, Then

Theorem 2: σ(N+M) = σ(N) + σ(M)

Proof:

σ(N+M) = σ(P(C, k)+P(D, k))
= σ(P((C+D), k) = P((C+D), e)
= P(C, e) + P(D, e) = σ(N) + σ(M)

Products of Polynomials

Again let N and M be two polynomials to the same base k, Then

Theorem 3: σ(NM) = σ(N)σ(M)

Proof:

σ(NM) = σ(P(C, k)P(D, k))
= P(C, e)P(D, e) = σ(N)σ(M)

Remainders

Let N be a polynomial of degree n and minimum power m, Then N may be expressed as

N = (k+(-e))M + R

where M is of degree one less than N and R is of a degree equal to m, the minimum power of N. Then:

Theorem 4: σ(N) = σ(R)

Proof:

σ(N) = σ((k+(-e))M+R)
= σ((k+(-e))σ(M) + σ(R)
but σ((k+(-e)))=0
therefore
σ(N) = σ(R)

R is of the form d₀k^m. The quantity d₀ is what usually would be called the remainder for division of N by (k+(-e)).

These proporties stated in the above theorems apply to any field including, of course, the field of complex numbers.

Conclusions

Polynomials in any mathematical field have the following properties. The coefficient sums of the sum and the product of two polynomials of the same base are the sum and product, respectively, of the coefficient sums of the polynomials. The difference between a polynomial and its coefficient sum is a multiple of its base less its multiplicative identity. The coefficient sum is equal to the remainder upon division by its base less its multiplicative identity.