To solve this problem, we will use the given polynomial identity step-by-step to find the correct Pythagorean triple when [tex]\( x = 9 \)[/tex] and [tex]\( y = 4 \)[/tex].
The polynomial identity given is:
[tex]\[ (x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2 \][/tex]
First, let's calculate each term separately using [tex]\( x = 9 \)[/tex] and [tex]\( y = 4 \)[/tex].
1. Calculate [tex]\( (x^2 + y^2)^2 \)[/tex]
[tex]\[
x^2 + y^2 = 9^2 + 4^2 = 81 + 16 = 97
\][/tex]
[tex]\[
(x^2 + y^2)^2 = 97^2 = 9409
\][/tex]
2. Calculate [tex]\( (x^2 - y^2)^2 \)[/tex]
[tex]\[
x^2 - y^2 = 9^2 - 4^2 = 81 - 16 = 65
\][/tex]
[tex]\[
(x^2 - y^2)^2 = 65^2 = 4225
\][/tex]
3. Calculate [tex]\( (2xy)^2 \)[/tex]
[tex]\[
2xy = 2 \cdot 9 \cdot 4 = 72
\][/tex]
[tex]\[
(2xy)^2 = 72^2 = 5184
\][/tex]
We now have the following terms:
[tex]\[
(x^2 + y^2)^2 = 9409
\][/tex]
[tex]\[
(x^2 - y^2)^2 = 4225
\][/tex]
[tex]\[
(2xy)^2 = 5184
\][/tex]
Checking the provided options, we compare our calculated values:
- Option a) [tex]\( 65, 72, 113 \)[/tex] does not match any of the terms.
- Option b) [tex]\( 16, 81, 97 \)[/tex] does not match.
- Option c) [tex]\( 36, 45, 97 \)[/tex] does not match.
- Option d) [tex]\( 65, 72, 97 \)[/tex] matches exactly with the given terms.
Hence, the correct Pythagorean triple when [tex]\( x = 9 \)[/tex] and [tex]\( y = 4 \)[/tex] is:
[tex]\[ \boxed{d) \ 65, 72, 97} \][/tex]
