To find the Pythagorean triple generated by the polynomial identity \( (x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2 \), we can start by calculating the values using \( x = 12 \) and \( y = 9 \).
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Calculate \( x^2 \) and \( y^2 \):
- \( x^2 = 12^2 = 144 \)
- \( y^2 = 9^2 = 81 \)
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Calculate \( x^2 + y^2 \):
- \( x^2 + y^2 = 144 + 81 = 225 \)
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Calculate \( (x^2 + y^2)^2 \):
- \( (x^2 + y^2)^2 = 225^2 = 50625 \)
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Calculate \( x^2 - y^2 \):
- \( x^2 - y^2 = 144 - 81 = 63 \)
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Calculate \( (x^2 - y^2)^2 \):
- \( (x^2 - y^2)^2 = 63^2 = 3969 \)
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Calculate \( 2xy \):
- \( 2xy = 2 \times 12 \times 9 = 216 \)
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Calculate \( (2xy)^2 \):
- \( (2xy)^2 = 216^2 = 46656 \)
Now verify the polynomial identity: \[ (x^2 - y^2)^2 + (2xy)^2 = 63^2 + 216^2 = 3969 + 46656 = 50625 \]
Since both sides equal \( 50625 \), we confirm the identity holds true. In a Pythagorean triple form \( a^2 + b^2 = c^2 \), we have:
- \( (x^2 + y^2) = 225 \) (this will be \( c^2 \))
- \( (x^2 - y^2) = 63 \) (this will be \( a \))
- \( 2xy = 216 \) (this will be \( b \))
Now we choose one of the values of the Pythagorean triple from the provided responses. The largest value \( 225 \) is the \( c^2 \) which is part of the Pythagorean triple.
Thus, one of the values of the Pythagorean triple is:
225
1 answer