What the heck is this paragraph saying?
In the first example, x + y + z, it's saying that if you swap, say, x and z, you get an different expression that has the same value.
IOW, x + y + z = z + y + x
In the second, bc + ca + ab, you can swap, say b for c, to get a new expression with the same value.
IOW, bc + ca + ab = cb + ba + ca.
bc = cb due to commutivity of multiplication, and ab = ba for the same reason. There's also some commutivity of addition going on, as ca + ab = ab + ca.
so its not talking about x, y, and origin symmetry?
No, its is not. It is talking about interchanging the letter variables themselves:
3x+ 3y is "symmetric" in this sense but x+ 2y is not.
and x+ 2y is not symmetric because the x and y coefficients are not equal?
The type of symmetry being discussed here reminds me of binomial expansions.
Notice that the coefficients aren't the only cause of symmetry!
what do they mean by "symmetry?" what will knowing symmetry of an expression help me with?
If you can perform some type of transformation to an expression or equation (without murdering it!), then there is a relational symmetry. In other words, the output of a function does not vary when the arrangement of any or all of the variables within the function are changed.
I'm an amateur, though, so I'm sure HoI or Mark can put it more succinctly or correct anything I've said that is ambiguous, imprecise, or simply incorrect.
You will encounter situations where things can be assumed or proven because of symmetrical relationships, and it will often save you a lot of work!
This video might be helpful to you:
Separate names with a comma.