In mathematics, and particularly in formal algebra, an indeterminate is a symbol that is treated as a variable, but does not stand for anything else but itself and is used as a placeholder in objects such as polynomials and formal power series.
So, how it is different from variable which is also used as a placeholder? What is the use of the indeterminate?
An indeterminate is a syntactic tool that is used to write down expressions in a certain way. If we are considering polynomials, for example, we may use $x$ as an indeterminate to write them down easily: $1 + 4x – x^4$. If $x$ is an indeterminate, then this is just a way that we can write down the list of coefficients $(1, 4, 0, 0, -1)$ more easily. We use the indeterminate $x$ for stylistic reasons and to ease computation; formally speaking, we are really talking about the list of coefficients.
On the other hand, a variable is used to represent an element of some set. Consider these examples: “Let $x$ be an integer.” “For all integers $n$, $n$ is even or odd.” “Let $x = 37$.” In all cases, the variable stands for an element of a set, whether it be a specific element or an arbitrary element. Variables are what we use to go about mathematical reasoning. Variables are fundamental to writing proofs; we cannot prove something about all elements of a set or any for all or there exists statement without utilizing variables.
Of course, usually when we use an indeterminate (at least, in the case of formal power series and polynomials) we want to emphasize a sort of analogy with variables, because it is possible to “plug in” a specific value for that indeterminate in the same way we can plug in a specific value to a variable. Therefore, in some ways indeterminates just stand for a particular kind of variable (but not the other way around).
$x^2 + 3x + 3$ is a polynomial which has $x$ as indeterminate. The $x$ doesn’t stand for any particular number, it’s just the indeterminate of the expression. You can ask various questions about the polynomial (e.g. “does it factor?”), but you can’t ask “what is $x$?”.
On the other hand, you might try to solve the equation $x^2 + 3x + 3 = 7$. Here $x$ is being used as a variable: it stands for an unknown number which makes the equation true when substituted for $x$.