Remarks

• In computations with polynomials the following properties of the two operations polynomial addition and multiplication are often tacitly used. They look quite straightforward and are easy to use in practice. But since we have constructed a new arithmetical structure, they do require proofs. Here is a list of the properties we mean (similar properties will reoccur in different contexts throughout Algebra Interactive; in fact arithmetic in the integers, and arithmetic modulo n satisfy such properties). For all polynomials a, b, c in R[X] the following properties hold.

  • a + b = b + a (commutativity of addition);
  • a · b = b · a (commutativity of multiplication);
  • (a + b) + c = a + (b + c) (associativity of addition);
  • (a · b) · c = a · (b · c) (associativity of multiplication);
  • a(b + c) = ab + ac (distributivity of multiplication over addition);
  • If a = a₀ + a₁X + ··· + a_mX^m then its (unique) opposite, i.e., the polynomial b such that a + b = 0 (the constant polynomial 0), is the polynomial
    -a₀ - a₁X - ··· - a_mX^m;

    The opposite of a is denoted by -a.

  • a + 0 = 0 + a = a;
  • a · 1 = 1 · a = a.

  To prove these properties, one computes left-hand and right-hand sides in each case using the definitions and compares coefficients. These are easy checks so we restrict ourselves to proving the second item. Let a = a₀ + a₁X + ··· + a_mX^m and b = b₀ + b₁X + ··· + b_mX^m (the implicit assumption that both polynomials end with a term of the form coefficient times Xⁿ is harmless). Then the coefficient of X^k in the product ab is

  a₀b_k + a₁b_{k - 1} + ··· + a_kb₀,

  whereas the coefficient of X^k in the product ba is

  b₀a_k + b₁a_{k - 1} + ··· + b_ka₀.

  It is easily seen that these two coefficients are equal. In fact, you are using commutativity for addition and multiplication of elements from R to verify the equality of the two coefficients!

• On the set R[X] we can also define another product *:

  ( _k a_k X ^k ) * ( _k b_k X ^k ) = _k (a_k b_k )X ^k .

  This product is called the Hadamard product. In this context the notation with the indeterminate X is not so practical.

• For a positive integer m and a polynomial a, we mean by a^m of course the product of m factors a:
  a · a ··· a.

  a⁰ is by definition 1.

• In practical computations you hardly need the explicit formula for the product of two polynomials. Check in examples that multiplying two polynomials is very much like ordinary arithmetic, with the additional rule that you replace products X^m · Xⁿ by X^{m + n}.