Question

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

Answer 1

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**