Use the polynomial identity (x^2+y2)^2=(x^2−y2)^2+(2xy)^2 to generate a Pythagorean triple when x equals 7 and y equals 3. Which of the following is one of the values of the Pythagorean triple?(1 point)
1. 40
2. 21
3. 1,764
4. 1.600
To generate a Pythagorean triple, we need to find values of x and y that satisfy the equation (x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2.
Let x = 7 and y = 3.
Substituting these values into the equation, we get:
(7^2 + 3^2)^2 = (7^2 - 3^2)^2 + (2*7*3)^2
(49 + 9)^2 = (49 - 9)^2 + (42)^2
(58)^2 = (40)^2 + (42)^2
3364 = 1600 + 1764
So, one of the values of the Pythagorean triple is 1,764.
Therefore, the correct answer is 3. 1,764.
To generate a Pythagorean triple using the polynomial identity (x^2+y^2)^2=(x^2−y^2)^2+(2xy)^2, we substitute x = 7 and y = 3 into the equation.
(x^2+y^2)^2 = (7^2+3^2)^2
= (49+9)^2
= 58^2
= 3364
(x^2−y^2)^2+(2xy)^2 = (7^2-3^2)^2+(2*7*3)^2
= (49-9)^2+(14)^2
= 40^2+196
= 1600+196
= 1796
Now, we have two values: 3364 and 1796. To find the Pythagorean triple, we need to take the square root of both values.
√(3364) = 58
√(1796) ≈ 42.38
Among the given options, the closest value to 42.38 is 40. Therefore, the correct answer is:
1. 40
To generate a Pythagorean triple, we need to substitute the values of x and y into the polynomial identity and solve for the resulting expression.
Let's substitute x = 7 and y = 3 into the polynomial identity:
(7^2 + 3^2)^2 = (7^2 - 3^2)^2 + (2 * 7 * 3)^2
Simplifying:
(49 + 9)^2 = (49 - 9)^2 + (14)^2
(58)^2 = (40)^2 + (14)^2
3364 = 1600 + 196
3364 = 1796
This equation is not true, so the values of x = 7 and y = 3 do not generate a Pythagorean triple.
Therefore, none of the provided answer choices (1. 40, 2. 21, 3. 1,764, 4. 1,600) is the correct value for the Pythagorean triple.