use the polynomial identity (x^2+y^2=(x^2-y^2+(2xy)^2 to generate a pythagorean triple when x equals 7 and y equals 4 write the pythagorean triple in assecnding order

Answer 1

To generate a Pythagorean triple using the polynomial identity \( x^2 + y^2 = (x^2 - y^2) + (2xy)^2 \), we'll use \( x = 7 \) and \( y = 4 \).

1. Calculate \( x^2 \) and \( y^2 \): \[ x^2 = 7^2 = 49 \] \[ y^2 = 4^2 = 16 \]

2. Calculate \( x^2 + y^2 \): \[ x^2 + y^2 = 49 + 16 = 65 \]

3. Calculate \( x^2 - y^2 \): \[ x^2 - y^2 = 49 - 16 = 33 \]

4. Calculate \( 2xy \): \[ 2xy = 2 \times 7 \times 4 = 56 \]

5. Now, we can write the Pythagorean triple as follows:

   • The three parts of the triple are \( x^2 - y^2 \), \( 2xy \), and \( x^2 + y^2 \).
   • Thus, the parts are \( 33 \), \( 56 \), and \( 65 \).

6. Sort the values in ascending order: \[ 33, 56, 65 \]

Therefore, the Pythagorean triple generated is \( (33, 56, 65) \).