Question
Use the polynomial identity (x^2+y^2)^2=(x^2−y^2)^2+(2xy)^2
to generate a Pythagorean triple when x equals 7 and y equals 4. Write the Pythagorean triple in ascending order.
Question
Use the polynomial identity (x2+y2)2=(x2−y2)2+(2xy)2
to generate a Pythagorean triple when x equals 7 and y equals 4. Write the Pythagorean triple in ascending order.
To generate a Pythagorean triple using the given polynomial identity, we can substitute x = 7 and y = 4.
Substituting x = 7 and y = 4 into the polynomial identity, we get:
(7^2 + 4^2)^2 = (7^2 - 4^2)^2 + (2 * 7 * 4)^2
(49 + 16)^2 = (49 - 16)^2 + (56)^2
(65)^2 = (33)^2 + (56)^2
4225 = 1089 + 3136
The equation 4225 = 4225 is true.
Therefore, by substituting x = 7 and y = 4 into the polynomial identity, we have generated the Pythagorean triple (33, 56, 65) in ascending order.
To generate a Pythagorean triple using the given polynomial identity, we can substitute x = 7 and y = 4 into the equation.
Substituting the values, we have:
(7^2 + 4^2)^2 = (7^2 - 4^2)^2 + (2 * 7 * 4)^2
Simplifying the equation:
(49 + 16)^2 = (49 - 16)^2 + (14)^2
(65)^2 = (33)^2 + (14)^2
4225 = 1089 + 196
4225 = 1285
Since this equation is not true, it means that (7, 4) does not form a Pythagorean triple.
To generate a Pythagorean triple using the given polynomial identity, we substitute x = 7 and y = 4 into the equation:
(x^2 + y^2)^2 = (x^2 − y^2)^2 + (2xy)^2
(7^2 + 4^2)^2 = (7^2 − 4^2)^2 + (2 * 7 * 4)^2
(49 + 16)^2 = (49 − 16)^2 + (2 * 7 * 4)^2
(65)^2 = (33)^2 + (56)^2
Now, we have a Pythagorean triple in the form of (a, b, c), where a = 33, b = 56, and c = 65. The Pythagorean theorem states that c^2 = a^2 + b^2. So, in ascending order, the Pythagorean triple is:
(33, 56, 65)