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 \( y = 4 \).
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Calculate \( x^2 \) and \( y^2 \): \[ x^2 = 7^2 = 49 \] \[ y^2 = 4^2 = 16 \]
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Calculate \( x^2 + y^2 \): \[ x^2 + y^2 = 49 + 16 = 65 \]
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Calculate \( x^2 - y^2 \): \[ x^2 - y^2 = 49 - 16 = 33 \]
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Calculate \( 2xy \): \[ 2xy = 2 \times 7 \times 4 = 56 \]
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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 \( 65 \).
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Sort the values in ascending order: \[ 33, 56, 65 \]
Therefore, the Pythagorean triple generated is \( (33, 56, 65) \).