Use the polynomial identity (x^2+y^2)^2=(x^2−y^2)^2+(2xy)^2 to generate a Pythagorean triple when x equals 11 and y equals 4. Which of the following is one of the values of the Pythagorean triple? (1 point)
a. 137
b. 225
c. 44
d. 49
To use the polynomial identity to generate a Pythagorean triple, we substitute x = 11 and y = 4 into the equation.
(x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2
(11^2 + 4^2)^2 = (11^2 - 4^2)^2 + (2 * 11 * 4)^2
(121 + 16)^2 = (121 - 16)^2 + (88)^2
(137)^2 = (105)^2 + (88)^2
Taking the square root of both sides:
137 = sqrt((105)^2 + (88)^2)
Now, we can see that the value 137 is one of the values of the Pythagorean triple. Therefore, the answer is:
a. 137
To generate a Pythagorean triple using the polynomial identity (x^2+y^2)^2=(x^2−y^2)^2+(2xy)^2, we can substitute x = 11 and y = 4 into the equation:
(11^2 + 4^2)^2 = (11^2 - 4^2)^2 + (2*11*4)^2
Simplifying this equation gives:
(121 + 16)^2 = (121 - 16)^2 + (88)^2
(137)^2 = (105)^2 + (88)^2
Comparing this equation to the Pythagorean theorem a^2 + b^2 = c^2, we can see that the values of a, b, and c are:
a = 105
b = 88
c = 137
Therefore, the correct answer is a. 137.
To generate a Pythagorean triple using the given polynomial identity, we can substitute x = 11 and y = 4 into the equation and solve for the three sides of the triangle.
Using (x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2
Substituting x = 11 and y = 4:
(11^2 + 4^2)^2 = (11^2 - 4^2)^2 + (2 * 11 * 4)^2
Simplifying:
(121 + 16)^2 = (121 - 16)^2 + (88)^2
(137)^2 = (105)^2 + (88)^2
To find one of the values of the Pythagorean triple, we can calculate the square root of each side:
√[(137)^2] = √[(105)^2 + (88)^2]
137 = √[(105)^2 + (88)^2]
Checking the answer options:
a. 137: This matches the calculated value and is a possible answer.
b. 225: This does not match the calculated value.
c. 44: This does not match the calculated value.
d. 49: This does not match the calculated value.
Therefore, the correct answer is a. 137.