Question

Jamar wants to generate a Pythagorean triple when x equals 12 and y equals 9 using the polynomial identity (x2+y2)2=(x2−y2)2+(2xy)2 . Which of the following is one of the values of the Pythagorean triple?(1 point) Responses 108 108 441 441 3,969 3,969 225

Answer 1

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 \(y^2\):

1. Calculate \(x^2\) and \(y^2\):
\[
x^2 = 12^2 = 144
\]
\[
y^2 = 9^2 = 81
\]

2. Now, we can calculate \(x^2 + y^2\):
\[
x^2 + y^2 = 144 + 81 = 225
\]

3. Next, we find \((x^2 + y^2)^2\):
\[
(x^2 + y^2)^2 = 225^2 = 50625
\]

4. Now, let's calculate the two components of the Pythagorean triple using the identities:
- \(x^2 - y^2\):
\[
x^2 - y^2 = 144 - 81 = 63
\]
- \(2xy\):
\[
2xy = 2 \cdot 12 \cdot 9 = 216
\]

5. We can then verify the identity:
- Compute \((x^2 - y^2)^2\):
\[
(63)^2 = 3969
\]
- Compute \((2xy)^2\):
\[
(216)^2 = 46656
\]

6. 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**.