Symmetric polynomial

Okay, imagine you have some toys. Let's say you have three toy cars: a red one, a blue one, and a yellow one. Now, let's say you want to play a game where you line them up in a row. You can do this in lots of different ways. For example, you could put the red car first, then the blue car, then the yellow car. Or you could put the blue car first, then the red car, then the yellow car. These are called arrangements of your cars.

A symmetric polynomial is kind of like an arrangement of toys, but with math instead. Instead of toy cars, you have variables and numbers. A variable is like a blank space that can be filled in with a number. For example, if you have a variable "x," you can put in different numbers for x, like 2 or 3 or 100.

A polynomial is just a bunch of these variables and numbers added together and multiplied by each other. For example, x^2 + 3x + 1 is a polynomial (it has a variable x raised to the power of 2, another x, and a number 1 added together).

A symmetric polynomial is a special kind of polynomial where if you change the order of the variables, the polynomial stays the same. Just like how if you change the order of your toy cars, the arrangement is still the same. For example, if you have a symmetric polynomial with variables x, y, and z, then it's the same polynomial whether you write it as (x+y+z)^2 or (x+z+y)^2 or (z+y+x)^2. They all have the same terms added and multiplied together.

So, a symmetric polynomial is like a special kind of math toy arrangement where the order doesn't matter.