100 Points!!! Determine if the equation (2x²+y²)²=(2x²-y²)²+(2xy√2)² is a polynomial identity. Photo attached. Please show as much work as possible. Thank you!

Question

asked May 5, 2024 225k views

1 Answer

Morten Olsen · Answer 1 · 2024-05-09T05:35:35+0000

Answer:

Yes, it is a polynomial identity (work below).

Explanation:

Let's start by expanding the left side of the equation.

$(2x^2+y^2)^2=\\4x^4+y^4+4x^2y^2$

Now, let's expand the second side.

$(2x^2-y^2)^2+(2xy√(2) )^2=\\4x^4+y^4-4x^2y^2+4x^2y^2(2)=\\4x^4+y^4-4x^2y^2+8x^2y^2$

There are no like terms on the left side, so let's combine the like terms on the right side.

$4x^4+y^4-4x^2y^2+8x^2y^2=\\4x^4+y^4+4x^2y^2$

Now, let's check if the left and right sides are equal:

4x^4+y^4+4x^2y^2= 4x^2+y^4+4x^2y^2

Both sides are the same. Thus, the equation is a polynomial identity.