Symmetric polynomials

Symmetric polynomials are like toys that you can play with to make them look the same from different angles. Imagine you have a toy car with red, blue, and green blocks. You can arrange the blocks in different orders to make a pretty design.

A symmetric polynomial is like a toy that has different variables instead of colored blocks. You can arrange these variables in different ways to make a polynomial. For example, the variables in the polynomial x+y could be rearranged as y+x. These two polynomials look different, but they actually mean the same thing.

Now imagine you have a set of rules that tell you how to rearrange the variables to make the polynomial look the same no matter what order you put them in. These rules are called symmetries. You can apply these symmetries to your toy polynomial to make it look the same from different angles.

A symmetric polynomial is a polynomial that looks the same no matter how you rearrange its variables using these symmetries. These types of polynomials are important in math because they have special properties that make them easier to work with. In fact, they often have solutions that can be found using less complicated methods than non-symmetric polynomials.