Intuition behind polynomial multiplication I noticed that multiplication of polynomial behaves as convolution of their coefficients. And this arises a question for me what is the intuition behind defining multiplication like this? Why we don’t define polynomial multiplication Components-wise instead? Like why not define multiplication like this $$(x^3+2x^2+1)(4x^2+2)=(0)x^3+(8)x^2 + (2)$$?
 A: Polynomial multiplication is using the usual definition of $\times$ that distributes over $+$. Using distributive property many times over gets the convolution. Component wise multiplication is often useful in data, but it’s just not the usual multiplication that you have from either substituting numbers for variables or thinking of $x$ as a formal variable.
A: With the coefficient-wise product, the evaluation map would not be a ring homomorphism. Many people think of polynomials as functions, so that would be a problem.
I know of an application of coefficient-wise multiplication (Hadamard product) in combinatorics, in the field of partially ordered sets. The paper I would recommend is by Wilhelm Plesken, "Counting with Groups and Rings" 1982, Journal fuel die reine und angewandte Mathematik.
A: Polynomials are functions.
But what are functions? Functions are where, if you input one number, you get a different number.
Consider your example, $(x^3 + 2x^2 +1)(4x^2 + 2)$. Here, we can use the distributive property, which is used for sums of numbers. But remember, polynomials are made of power functions, so polynomials are sums of numbers.
$$(x^3 + 2x^2 + 1)(4x^2) + (x^3 + 2x^2 + 1)(2)$$
Then, since multiplication is communitive, we can do
$$((x^3)(4x^2) + (2x^2)(4x^2) + 1(4x^2)) + ((x^3)(2) + (2x^2)(2) + 1(2))$$
Which is what polynomial multiplication is.
You make a very interesting connection between polynomial multiplication and convolutions. In fact, some Python libraries use polynomial multiplication to calculate convolutions. The video below by 3Blue1Brown gives more information about his. Go to 13:40 for that.
https://www.youtube.com/watch?v=KuXjwB4LzSA&t=1197s
