What is "multiplying polynomials"?

Let's have two polynomials, f and g, written in this way:

f=(a0,a1,a2,.....,ak,…an,...), g=(b0,b1,b2,…,bm,....)

where a0, a1, a2....an, b0, b1,....bn are coefficients of these two polynomials.

We can write the polynomials in this way also:

f(x)=a0X^n + a1X^(n-1) + .......+anX^0

g(x)=b0X^m + b1X^(m-1) + .......+bmX^0

These two polinoms have a finite number of terms, different from 0 value. We can define on the set of complex numbers the following algebraical operations: addition and multiplication.

f+g=(a0+b0,a1+b1,a2+b2,...)

fg=(c0,c1,c2,...), (2)

where

c0=a0*b0,

c1=a0*b1+a1*b0,

c2=a0*b2+a1*b1+a2*b0,

ar=a0*br+a1*br-1+a2*br-2+...+ar*b0= ai*br-i= ai*bj

The element f+g=(a0+b0,a1+b1,....) is called the sum between f and g and the operation is called addition.

The element f*g=(c0,c1,c2,....) is called the product between f and g,and the operation is called multiplication.

For example:

If f=(-1,2,3,-5,0,0,..) and g=(1,0,-1,0,...), then their sum is f+g=(0,2,2,-5,0,0,...), and their product is f*g=(-1,2,4,-7,-3,5,0....).

The properties of polynomial multiplication:

1. Commutation between the factors of mutiplying does not change the result:

f*g=g*f

2. The multiplication is associative.

(f*g)*h=f*(g*h)

3. The polynomial 1=(1,0,0,...) is neutral element for multiplication.

f*1=1*f=f

4. The mutiplication is distributive with respect to addition.

f*(g+h)=f*g+g*h
(f+g)*h=f*h+g*h

5. f*g=f*h and fis non-zero polynomial ,then we can simplify with f and the result is g=h