Coefficients are added together because they are like terms, this can be proven with the distributive property. For example, x(2x+x)=2x^2+x^2=3x^2.
The commutative property of addition and the associative property demonstrate this.
The word "commutative" comes from "commute" or "move around", so the Commutative Property is the one that refers to moving values around.
The associative property states that you can add or multiply regardless of how the numbers are grouped.