Symmetric power
Symmetric power is when we take a number and multiply it by itself. But instead of just doing it once, we do it a certain number of times, depending on what we want.
For example, let's say we want to find the symmetric power of the number 2. If we want to take it to the power of 2, we would write it as 2^2, which is 2 multiplied by 2, and equals 4. If we wanted to take it to the power of 3, we would write it as 2^3, which is 2 multiplied by 2 multiplied by 2, and equals 8.
Now, let's say we want to find the symmetric power of a polynomial. A polynomial is just a bunch of numbers and variables like x, y, or z. To find the symmetric power of a polynomial, we do the same thing as before, but now we multiply the polynomial by itself the number of times we want.
For example, if we want to find the symmetric power of the polynomial (x + y), and we want to take it to the power of 2, we write it as (x + y)^2, which means (x + y) multiplied by itself, and equals x^2 + 2xy + y^2. If we wanted to take it to the power of 3, we would write it as (x + y)^3, which means (x + y) multiplied by itself three times, and equals x^3 + 3x^2y + 3xy^2 + y^3.
The more times we multiply the polynomial by itself, the longer and more complicated the answer becomes. But that's basically all there is to the symmetric power!
For example, let's say we want to find the symmetric power of the number 2. If we want to take it to the power of 2, we would write it as 2^2, which is 2 multiplied by 2, and equals 4. If we wanted to take it to the power of 3, we would write it as 2^3, which is 2 multiplied by 2 multiplied by 2, and equals 8.
Now, let's say we want to find the symmetric power of a polynomial. A polynomial is just a bunch of numbers and variables like x, y, or z. To find the symmetric power of a polynomial, we do the same thing as before, but now we multiply the polynomial by itself the number of times we want.
For example, if we want to find the symmetric power of the polynomial (x + y), and we want to take it to the power of 2, we write it as (x + y)^2, which means (x + y) multiplied by itself, and equals x^2 + 2xy + y^2. If we wanted to take it to the power of 3, we would write it as (x + y)^3, which means (x + y) multiplied by itself three times, and equals x^3 + 3x^2y + 3xy^2 + y^3.
The more times we multiply the polynomial by itself, the longer and more complicated the answer becomes. But that's basically all there is to the symmetric power!