Polynomials as a Linear Combination

Question

I have read that the set of all Polynomials Pⁿ are also a set of vector spaces. And the explain I read, said that apart from following all the properties of a vector space (identity, communtativity, etc.), it is also the linear combination of scalars (coefficients) and vectors (variables).

But, as far as I know Polynomials can be of any degree and not necessarily be linear. So, are Polynomials really a linear combination of scalars and vectors and if so, how?

I think your confusion is due to the fact that the word "linear" is used in more than one way -- linear polynomials as being 1st degree polynomials (or at most 1st degree polynomials; author usage/definitions vary on this particular issue) and polynomials being linear combinations of powers of $x$ (i.e. the polynomial $3x^4 - 5x^3 + x^2 - 2x + 3$ is a linear combination of "vectors" in the set ${1,,x,,x^2,,x^3,,x^4}).$ — Dave L. Renfro, Jun 26 '24 at 06:26
@Dave L. Renfro, Thank for the point, I really didn't know the other usage of linear. P.S.- I was truly thinking linear in terms of degree. — inc0gnizant, Jun 26 '24 at 11:18
FYI, there are several (English) words much worse than "linear" for having multiple mathematical meanings. My guess for the worst offender is "normal" -- see Universal meaning of 'normal' (note also normal number and normal topological space usages). Another offender much worse than "linear" are various versions of "modular" -- What is the origin of the term "modular" in different areas? — Dave L. Renfro, Jun 26 '24 at 14:39

Polynomials as a Linear Combination

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