Question

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 \( (33, 56, 65) \).