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

Answers

There are no human answers yet.
Answered by GPT-4o mini
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**.

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