I've read the definition of a polynomial on Wikipedia, and got quite a different understanding from what was explained to me in a Khan Academy video.
In the Khan Academy video it says that:
$6$
is a polynomial, specifically a monomial because it's the same as:
$6x^0$
I was wondering if this was true, and also whether:
$6 + 1$
is a binomial or
6 + 15 - 2
is a trinomial
The Wikipedia articles defines a polynomial as an:
...expression consisting of variables (also called indeterminates) and coefficients...
And gives examples of:
$x^2 − 4x + 7$ and $x3 + 2xyz2 − yz + 1$
Something like I've shown higher up in my question, such as adding or subtracting 3 integers doesn't look like your typical polynomial. If 6 is a monomial and therefore a polynomial, then also 6 + 15 - 2 is a trinomial and polynomial, even if there are no explicitly written variables or indeterminates, as Wikipedia refers to them?
Regarding the one variable case, if you want to be extra rigorous, one can define polynomials as lists of numbers $(a_0,a_1, \dots)$ such that the $a_i$ eventually start to be all zero. This is just a fancy way of saying that $3$ can be defined as $(3,0,0,\dots)$ and $3X+X^5$ as $(0,3,0,0,0,1,0,0,\dots)$, i.e. grouping all the coefficients that belong to a same power together.
What this means is that even if we informally have different terms with the same variable, at the time of assessing certain properties of a polynomial, one should write it into this "canonical" form. For example, we could write $7$ as $6+1$ or $X^2 + 4X$ as $X^2 + X + 3X$ or even $X(X+4)$. But $6+1$ is a monomial, because after regrouping, it corresponds to $(7,0,0, \dots)$. In the same way, $X^2 + X + 3X$ consists of two monomials, because it corresponds to $(0,4,1,0,0,\dots)$.