Which letter is the coefficient here? $x(a+b)$ and $ax+bx$ have the same meaning, only $ax+bx$ is an expanded version.  
In $x(a+b)$, it seems like $x$ is the coefficient and $a$ and $b$ are variables, while in $ax+bx$, it seems like $x$ is the variable and $a$ and $b$ are coefficients.  
How can they be different if the expressions are the same, only in a different form?
 A: As The Chaz 2.0 says, convention leads us to a particular answer: this is very probably intended to be treated as a polynomial in $x$ alone. In that case, "the coefficient of $a$" doesn't make sense, and the coefficient of $x$ is $a+b$ (not "$a$ and $b$", or something like that).

The expression strongly suggests the interpretation above, but if someone wanted to be mean/confusing, there are some other options:


*

*If it's a polynomial in $y$ alone, then $ax+bx$ is the constant term. Perhaps you might call that the coefficient of $y^0$, but there are certainly no other non-zero coefficients.

*If it's a polynomial in $x$ (and other non-$a$, non-$b$ variables), then the coefficient of $x$ is $a+b$, and there aren't really other coefficients (the constant term is $0$).

*If it's a polynomial in $a$ (and not $b$ nor $x$, but maybe other variables), then the coefficient of $a$ is $x$ and the constant term is $bx$. The case of it being a polynomial in $b$ but not the other two variables is analogous.

*If it's a polynomial in $a$ and $x$ (and perhaps other non-$b$ variables), then the coefficient of the $ax$ term is $1$, and the coefficient of the $x$ term is $b$. The "polynomial in $b$ and $x$ but not $a$" case is analogous.

*If it's a polynomial in $a$ and $b$ and $x$ (ans perhaps other variables), then the coefficient of the $ax$ term is $1$, and the coefficient of the $bx$ term is $1$. There are no other non-zero coefficients.

