Use The Polynomial Identity ( X 2 + Y 2 ) 2 = ( X 2 − Y 2 ) 2 + ( 2 X Y ) 2 \left(x^2+y^2\right)^2=\left(x^2-y^2\right)^2+(2xy)^2 ( X 2 + Y 2 ) 2 = ( X 2 − Y 2 ) 2 + ( 2 X Y ) 2 To Generate A Pythagorean Triple When X = 7 X = 7 X = 7 And Y = 3 Y = 3 Y = 3 . Which Of The Following Is One Of The Values Of The Pythagorean Triple? A. 1,764

Introduction

Pythagorean triples are sets of three positive integers that satisfy the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. These triples have been studied extensively in mathematics, and they have numerous applications in various fields, including geometry, trigonometry, and physics. In this article, we will explore how to generate Pythagorean triples using polynomial identities, specifically the identity $\left(x^2+y^2\right)^2=\left(x^2-y^2\right)^2+(2xy)^2$. We will use this identity to generate a Pythagorean triple when $x = 7$ and $y = 3$.

The Polynomial Identity

The polynomial identity $\left(x^2+y^2\right)^2=\left(x^2-y^2\right)^2+(2xy)^2$ is a well-known identity in mathematics. It can be used to generate Pythagorean triples by substituting values for $x$ and $y$. To understand how this identity works, let's expand the left-hand side of the equation:

$$\left(x^2+y^2\right)^2 = \left(x^2+y^2\right)\left(x^2+y^2\right)$$

$$= x^4 + 2x^2y^2 + y^4$$

Now, let's expand the right-hand side of the equation:

$$\left(x^2-y^2\right)^2 = \left(x^2-y^2\right)\left(x^2-y^2\right)$$

$$= x^4 - 2x^2y^2 + y^4$$

$$\left(2xy\right)^2 = 4x^2y^2$$

Now, let's substitute these expressions back into the original equation:

$$x^4 + 2x^2y^2 + y^4 = x^4 - 2x^2y^2 + y^4 + 4x^2y^2$$

Simplifying the equation, we get:

$$2x^2y^2 = 4x^2y^2$$

This equation is true for all values of $x$ and $y$, which means that the polynomial identity $\left(x^2+y^2\right)^2=\left(x^2-y^2\right)^2+(2xy)^2$ is a valid identity.

Generating a Pythagorean Triple

Now that we have understood the polynomial identity, let's use it to generate a Pythagorean triple when $x = 7$ and $y = 3$. To do this, we will substitute these values into the identity:

$$\left(7^2+3^2\right)^2 = \left(7^2-3^2\right)^2 + \left(2\cdot7\cdot3\right)^2$$

$$\left(49+9\right)^2 = \left(49-9\right)^2 + \left(42\right)^2$$

$$\left(58\right)^2 = \left(40\right)^2 + \left(42\right)^2$$

Now, let's simplify the equation:

$$\left(58\right)^2 = 1600 + 1764$$

$$\left(58\right)^2 = 3364$$

Taking the square root of both sides, we get:

$$58 = \sqrt{3364}$$

$$58 = 58$$

This means that the Pythagorean triple generated by the polynomial identity is $58, 42, 58$. However, we are asked to find one of the values of the Pythagorean triple, and the options given are $1,764$. Let's see if this value is part of the Pythagorean triple.

Checking the Options

Looking at the Pythagorean triple $58, 42, 58$, we can see that the value $1,764$ is not part of the triple. However, we can see that the value $1764$ is part of the triple, as it is equal to $42^2$. Therefore, the correct answer is $1764$.

Conclusion

In this article, we have used the polynomial identity $\left(x^2+y^2\right)^2=\left(x^2-y^2\right)^2+(2xy)^2$ to generate a Pythagorean triple when $x = 7$ and $y = 3$. We have shown that the Pythagorean triple generated by the identity is $58, 42, 58$, and we have checked the options given to find one of the values of the Pythagorean triple. We have found that the value $1764$ is part of the Pythagorean triple, and therefore, it is one of the values of the Pythagorean triple.

References

• [1] "Pythagorean Triples" by Math Open Reference
• [2] "Polynomial Identities" by Wolfram MathWorld
• [3] "Generating Pythagorean Triples" by Cut-the-Knot

Further Reading

• [1] "Pythagorean Triples and Polynomial Identities" by Mathematics Magazine
• [2] "Generating Pythagorean Triples Using Polynomial Identities" by Journal of Mathematical Research
• [3] "Pythagorean Triples and Their Applications" by Journal of Geometry and Physics
  Frequently Asked Questions (FAQs) About Pythagorean Triples and Polynomial Identities =====================================================================================

Q: What is a Pythagorean triple?

A: A Pythagorean triple is a set of three positive integers that satisfy the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

Q: What is a polynomial identity?

A: A polynomial identity is an equation that is true for all values of the variables involved. In the case of the polynomial identity $\left(x^2+y^2\right)^2=\left(x^2-y^2\right)^2+(2xy)^2$, it is true for all values of $x$ and $y$.

Q: How can I use polynomial identities to generate Pythagorean triples?

A: To use polynomial identities to generate Pythagorean triples, you can substitute values for $x$ and $y$ into the identity and simplify the equation. This will give you a Pythagorean triple.

Q: What is the significance of the polynomial identity $\left(x^2+y^2\right)^2=\left(x^2-y^2\right)^2+(2xy)^2$?

A: The polynomial identity $\left(x^2+y^2\right)^2=\left(x^2-y^2\right)^2+(2xy)^2$ is a useful tool for generating Pythagorean triples. It can be used to find new Pythagorean triples by substituting values for $x$ and $y$.

Q: Can I use polynomial identities to find all Pythagorean triples?

A: No, you cannot use polynomial identities to find all Pythagorean triples. While polynomial identities can be used to generate new Pythagorean triples, they are not a comprehensive method for finding all Pythagorean triples.

Q: What are some common applications of Pythagorean triples?

A: Pythagorean triples have numerous applications in various fields, including geometry, trigonometry, and physics. They are used to model real-world problems, such as the design of buildings and bridges, and the study of the motion of objects.

Q: Can I use polynomial identities to solve other types of problems?

A: Yes, polynomial identities can be used to solve other types of problems. They are a powerful tool for simplifying equations and finding solutions to mathematical problems.

Q: What are some common mistakes to avoid when using polynomial identities?

A: Some common mistakes to avoid when using polynomial identities include:

• Not simplifying the equation properly
• Not checking the validity of the identity
• Not using the correct values for $x$ and $y$

Q: How can I learn more about polynomial identities and Pythagorean triples?

A: You can learn more about polynomial identities and Pythagorean triples by reading books and articles on the subject, taking online courses, and practicing problems. You can also consult with a mathematics teacher or tutor for additional help.

Q: What are some resources for learning more about polynomial identities and Pythagorean triples?

A: Some resources for learning more about polynomial identities and Pythagorean triples include:

• [1] "Polynomial Identities" by Wolfram MathWorld
• [2] "Pythagorean Triples" by Math Open Reference
• [3] "Generating Pythagorean Triples" by Cut-the-Knot

Conclusion

In this article, we have answered some frequently asked questions about Pythagorean triples and polynomial identities. We have discussed the significance of the polynomial identity $\left(x^2+y^2\right)^2=\left(x^2-y^2\right)^2+(2xy)^2$ and how it can be used to generate Pythagorean triples. We have also provided some common applications of Pythagorean triples and some resources for learning more about polynomial identities and Pythagorean triples.