Use the polynomial identity (x2+y2)2=(x2−y2)2+(2xy)2 to generate a Pythagorean triple when x equals 7 and y equals 3
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:
(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
3364 = 3364
Thus, we have generated the Pythagorean triple (3364, 1600, 1764) using the given values of x = 7 and y = 3.
Matilde wants to build a wooden planter in the shape of a right triangle for their garden. One side will be formed by an existing bench that is 96 inches long. To fence the other two sides, Matilde wants to use pieces of wood that are each precut to 12 inches, so each of the two undetermined side lengths will need to be in whole number of inches. Use the polynomial identity (x2+y2)2=(x2−y2)2+(2xy)2 to find one of the other possible side lengths, assuming 2xy=96 and that the two undetermined sides are of different lengths. Which of the following values could be one of the side lengths for their wooden planter?
To find one of the possible side lengths for the wooden planter, we can use the given equation from the polynomial identity: (x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2.
Let's substitute 2xy = 96 into the equation: (x^2 + y^2)^2 = (x^2 - y^2)^2 + (96)^2.
Simplifying the equation, we have: (x^2 + y^2)^2 - (x^2 - y^2)^2 = 96^2.
Expanding both sides of the equation: x^4 + 2x^2y^2 + y^4 - (x^4 - 2x^2y^2 + y^4) = 96^2.
Simplifying further, we get: 4x^2y^2 = 96^2.
Dividing both sides by 4: x^2y^2 = (96/2)^2.
Solving for xy: xy = 48.
Since the question states that the undetermined sides need to be of different lengths, we need to find possible factor pairs of 48.
The factor pairs of 48 are: (1, 48), (2, 24), (3, 16), (4, 12), (6, 8).
Thus, the possible values for xy are 1*48, 2*24, 3*16, 4*12, and 6*8, which are: 48, 48, 48, 48, and 48 respectively.
Therefore, one possible side length for the wooden planter is 48 inches.
Which of the following best summarizes the impact of Shays’s Rebellion?
Shays's Rebellion, which occurred in 1786-1787, was an armed uprising in Massachusetts led by farmers and Revolutionary War veterans protesting against high taxes, debt, and lack of economic relief. It had several impacts:
1. Highlighted weaknesses of the Articles of Confederation: Shays's Rebellion exposed the flaws and limitations of the government under the Articles of Confederation. The rebellion's successful defiance of state authority revealed the inability of the central government to address domestic unrest and maintain order.
2. Pushed for constitutional reform: The rebellion provided a catalyst for the need to revise the Articles of Confederation, leading to the Constitutional Convention in 1787. The violence and disorder caused by Shays's Rebellion heightened concerns about a weak central government and demonstrated the necessity for a stronger federal system.
3. Strengthened federal authority and central government: The rebellion contributed to the overall sentiment that a stronger central government was required to prevent similar uprisings and address economic issues. It played a role in shaping the Constitution, which created a more powerful federal government and endowed it with more authority to maintain order and regulate the economy.
4. Increased support for ratification of the Constitution: The rebellion and the subsequent unrest it caused swayed public opinion towards supporting ratification of the Constitution. Concerns over the stability of the nation and the ability to address issues effectively were heightened, leading many to support the new Constitution as a means to prevent future uprisings.
Overall, Shays's Rebellion exposed the weaknesses of the Articles of Confederation, prompted the call for a stronger central government, and contributed to the ratification of the Constitution. It helped set the stage for the creation of the United States' current federal system of government.
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 = 7 and y = 3.
Let's plug in the values:
LHS: (x^2 + y^2)^2 = (7^2 + 3^2)^2 = (49 + 9)^2 = 58^2
RHS: (x^2 - y^2)^2 + (2xy)^2 = (7^2 - 3^2)^2 + (2 * 7 * 3)^2 = (49 - 9)^2 + 42^2 = 40^2 + 42^2 = 1600 + 1764 = 3364
Since the LHS and RHS are equal, we have:
58^2 = 3364
Therefore, the Pythagorean triple when x = 7 and y = 3 is (58, 3364).
To generate a Pythagorean triple using the given polynomial identity (x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2, we will substitute x = 7 and y = 3 into the equation and simplify it to find a Pythagorean triple.
Step 1: Substitute the given values of x and y into the polynomial identity:
(7^2 + 3^2)^2 = (7^2 - 3^2)^2 + (2 × 7 × 3)^2
Simplifying further:
(49 + 9)^2 = (49 - 9)^2 + (14)^2
(58)^2 = (40)^2 + 14^2
Step 2: Evaluate the squares:
3364 = 1600 + 196
Step 3: Combine like terms:
3364 = 1796
Step 4: We can see that the equation is not balanced; therefore, there is no Pythagorean triple generated by the given values of x = 7 and y = 3 using the polynomial identity provided.
In general, the polynomial identity can be used to generate Pythagorean triples by picking suitable values for x and y. However, in this case, the given values do not produce a valid Pythagorean triple.