To generate a Pythagorean triple using the given polynomial identity \( (x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2 \), we can substitute \( x = 12 \) and \( y = 9 \) into the identity.
First, we calculate \( x^2 \) and \( y^2 \):
\[ x^2 = 12^2 = 144 \]
\[ y^2 = 9^2 = 81 \]
Next, we compute \( x^2 + y^2 \) and \( x^2 - y^2 \):
\[ x^2 + y^2 = 144 + 81 = 225 \]
\[ x^2 - y^2 = 144 - 81 = 63 \]
Next, compute \( 2xy \):
\[ 2xy = 2 \cdot 12 \cdot 9 = 216 \]
Now we can plug these values into the right side of the polynomial identity:
\[ (x^2 + y^2)^2 = 225^2 \]
Calculating \( 225^2 \):
\[ 225^2 = 50625 \]
For the right side of the polynomial identity, we calculate:
\[ (x^2 - y^2)^2 + (2xy)^2 = 63^2 + 216^2 \]
Now compute \( 63^2 \) and \( 216^2 \):
\[ 63^2 = 3969 \]
\[ 216^2 = 46656 \]
Adding these two results:
\[ 63^2 + 216^2 = 3969 + 46656 = 50625 \]
Since both sides match, we confirm the identity holds with these values. Next, the values in the Pythagorean triple generated from the values computed are:
- The values derived from \( x^2 + y^2 \) and \( 2xy \) include \( 225 \) and \( 216 \).
The options are:
A. 3,969
B. 225
C. 108
D. 441
Among these, the value \( 225 \) (which represents \( x^2 + y^2 \)) is one of the components of the Pythagorean triple. Thus, the answer is:
B. 225
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