Polynomial over a Number System

Polynomial: Let K be any number system and x a formal symbol (called an indeterminent or a variable), then an expression of the form a₀ + a₁x + a₂x² + … + a_ixⁱ + …, where a_i's are the elements of the number system K, and only finitely many of a_i's are non-zero is called a polynomial in x over K.
Alternatively, an expression of the form 
a₀ + a₁x + a₂x² + … + a_ixⁱ + … is said to be a polynomial over a number system K if  a_i ∈ K for all i = 0, 1, 2, ... and there exists a non-negative integer m such that  a_i = 0 for all i ≥ m.
The polynomials in x over number system K are usually denoted by the symbols f(x), g(x), h(x) etc.
Let f(x) = a₀ + a₁x + a₂x² + … + a_ixⁱ + … be a polynomial in x over the number system K, then
        (i) The expression a₀ , a₁x , a₂x² , … , a_ixⁱ , … are called the terms of the polynomial f(x).
       (ii) The numbers a₀ ,a₁, a₂, … , a_i, …  are called the coefficients of the polynomial f(x), and the number  a_i is called the coefficient of xⁱ.
       (iii) a₀ is called the constant term of the polynomial f(x).
       (iv) Let f(x) = a₀ + a₁x + a₂x² + … + a_nxⁿ + 0xⁿ⁺¹ + 0xⁿ⁺² + … + a_n ≠ 0, 
             then f(x) is said to be a polynomial of degree n,
             the coefficient a_n is called the  leading coefficient of f(x) and the term a_nxⁿ is called the leading term of f(x), if the leading coefficient is 1,then f(x) is called a monic polynomial.

Equality of two Polynomials:
Let f(x) = a₀ + a₁x + a₂x² + … + a_nxⁿ + ... and g(x) = b₀ + b₁x + b₂x² + … + b_mx^m + ... be two polynomials in x over the same number system K, then f(x), g(x) are said to be equal written as f(x) = g(x), iff a_i = b_i for all i ≥ 0
i.e., iff all their corresponding coefficients are equal.

The polynomials 5 - 7x² , 5 + 0x +(-7)x² + 0x³ are equal while the polynomials 3+x² , 3+x³ are not equal.

Degree of a Polynomial:
Let f(x) = a₀ + a₁x + a₂x² + … + a_nxⁿ + ... be a non-zero polynomial over a number system K. If a_n ≠ 0 and a_i = 0 for all i >n then n is called the degree of f(x), it is written as 
deg f(x)=n.

Note 1: The degree of zero polynomial is not defined.

Note 2: The degree of a non-zero polynomial is always a non-negative integer.

Note 3: The degree of a non-zero polynomial is the index of power of x in its leading term.

                     Operations on Polynomials:
Addition of Polynomials: Let  f(x) = a₀ + a₁x + a₂x² + … + a_ixⁱ + … and 
 g(x) = b₀ + b₁x + b₂x² + … + b_ixⁱ + … be two polynomials in (an indeterminate x) over a number system K, then polynomial c₀ + c₁x + c₂x² + … + c_ixⁱ + … where c_i = a_i + b_i for every i ≥ 0 is called the sum of the polynomials f(x) and g(x), it is denoted by f(x) + g(x).
Thus f(x) + g(x) = (a₀ + b₀) + (a₁ + b₁)x + ... + (a_i + b_i)xⁱ + ...
    i.e., the sum of two polynomials f(x)and g(x)is a polynomial in which the coefficient of xⁱ  is the sum of the soefficient of xⁱ  in f(x) and the coefficient of xⁱ  in g(x) for every i ≥ 0.

Subtraction of Polynomials: Let f(x), g(x) be two polynomials in x over a number system K, then the difference of  f(x) and g(x) is denoted by f(x) - g(x) and it is defined as f(x) - g(x) = f(x) + (-g(x)).
     Thus the difference of the polynomials f(x) and g(x) is equal to the sum of the polynomials  f(x) and -g(x).

Multiplication of Polynomials: 
Let  f(x) = a₀ + a₁x + a₂x² + … + a_ixⁱ + … and 
      g(x) = b₀ + b₁x + b₂x² + … + b_ixⁱ + … be two polynomials in (an indeterminate x) over a number system K, then polynomial c₀ + c₁x + c₂x² + … + c_ixⁱ + … where c_i = a₀b_i + a₁b_i-1 + ... + a_rb_i-r + ... + a_ib₀ =  ∑a_rb_s
such that r + s = i and i and r, s are non-negative integers.
i.e., 0 ≤ r ≤ i, 0 ≤ s ≤ i is called the product of the polynomials f(x) and g(x), it is denoted by f(x)g(x).
     Thus if f(x) = a₀ + a₁x + a₂x² + … + a_ixⁱ + …  and 
                 g(x) = b₀ + b₁x + b₂x² + … + b_ixⁱ + …, then
           f(x)g(x) = a₀b₀ + (a₀b₁ + a₁b₀)x + (a₀b₂ + a₁b₁ + a₂b₀)x² + ... + (a₀b₁ + a₁b_i-1 + ... +a_ib₀)xⁱ + ...