Final answer:
The Pythagorean Identity (x²-y²)²+(2xy)²=(x²+y²)² is a polynomial identity commonly used to generate Pythagorean Triples, which are sets of three positive integers that satisfy the Pythagorean theorem. This equation relates the squares of the sides of a right triangle.
Step-by-step explanation:
The equation provided, (x²-y²)²+(2xy)²=(x²+y²)², is a polynomial identity called the Pythagorean Identity. This identity is commonly used to generate Pythagorean Triples, which are sets of three positive integers that satisfy the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
To understand how this identity creates Pythagorean Triples, we can consider an example. Let's say we have values for x and y, such as x = 3 and y = 4. We can substitute these values into the equation:
(3²-4²)²+(2*3*4)²=(3²+4²)²
(9-16)²+(24)²=(9+16)²
(-7)²+(24)²=(25)²
49+576=625
625=625
As we can see, the left side of the equation simplifies to the right side, confirming that the values x = 3 and y = 4 create a Pythagorean Triple.