To generate a Pythagorean triple using the polynomial identity \((x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2\) with \(x = 12\) and \(y = 9\), we'll first compute \(x^2\) and \(y^2\):
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Calculate \(x^2\) and \(y^2\): \[ x^2 = 12^2 = 144 \] \[ y^2 = 9^2 = 81 \]
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Now, we can calculate \(x^2 + y^2\): \[ x^2 + y^2 = 144 + 81 = 225 \]
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Next, we find \((x^2 + y^2)^2\): \[ (x^2 + y^2)^2 = 225^2 = 50625 \]
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Now, let's calculate the two components of the Pythagorean triple using the identities:
- \(x^2 - y^2\): \[ x^2 - y^2 = 144 - 81 = 63 \]
- \(2xy\): \[ 2xy = 2 \cdot 12 \cdot 9 = 216 \]
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We can then verify the identity:
- Compute \((x^2 - y^2)^2\): \[ (63)^2 = 3969 \]
- Compute \((2xy)^2\): \[ (216)^2 = 46656 \]
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Now, let's confirm the identity: \[ 63^2 + (2 \cdot 12 \cdot 9)^2 = 3969 + 46656 = 50625 \]
This checks out, verifying the use of the polynomial identity given \(x\) and \(y\).
Among the provided options (108, 441, 3969, 225), we see that:
- 3969 is one of the values of the Pythagorean triple.
Thus, the answer is:
3969.
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