Question

CRITICAL THINKING Recall that a Pythagorean triple is a set of positive integers \(a, b\), and \(c\) such that \(a^2+b^2=c^2\). The numbers 3,4 , and 5 form a Pythagorean triple because \(3^2+4^2=5^2\). You can use the polynomial identity \(\left(x^2-y^2\right)^2+(2 x y)^2=\left(x^2+y^2\right)^2\) to generate other Pythagorean triples. a. Prove the polynomial identity is true by showing that the simplified expressions for the left and right sides are the same. b. Use the identity to generate the Pythagorean triple when \(x=6\) and \(y=5\). c. Verify that your answer in part (b) satisfies \(a^2+b^2=c^2\)

Short answer

By substituting the values into the identity, we get the Pythagorean triple as 11, 60 and 61. On verification, we find that they satisfy the equation \(a^2+b^2=c^2\).

Step-by-step solution

Prove the Polynomial Identity

Start by simplifying the left side and the right side of the equation separately. The left side is \(\left(x^2-y^2\right)^2+(2 x y)^2\) and the right side is \(\left(x^2+y^2\right)^2\).

Generate the Pythagorean Triple

Use the identity and substitute \(x=6\) and \(y=5\) into the simplified equation. This will give values for \(a, b\), and \(c\) that should form a Pythagorean triple.

Verify the Pythagorean Triple

Take the numbers obtained from previous step and check if they satisfy \(a^2+b^2=c^2\) equation. If they do, then we have successfully used the identity to generate a valid Pythagorean triple.

Key concepts

Understanding Polynomial Identity

A polynomial identity is a mathematical equation that holds true for all values of variables involved. In this exercise, we explore the identity \((x^2-y^2)^2+(2xy)^2=(x^2+y^2)^2\). This identity is particularly useful because it shows a relationship between different forms of squared polynomials.

To prove this identity, simplify both sides separately. For the left side, expand \((x^2-y^2)^2\) and \((2xy)^2\), then combine the results. Doing the same for the right side, you'll find that the expanded expressions are equivalent. This confirms that the equation is indeed an identity.

Using such identities helps us transform complex algebraic expressions into simpler or more useful forms, which is key in generating Pythagorean triples.

Exploring Integer Solutions

Integer solutions involve finding whole number values that satisfy an equation. Pythagorean triples are an example of integer solutions where \(a^2 + b^2 = c^2\). Here, the challenge is to use the given polynomial identity to generate these integer solutions.

By substituting specific integer values for \(x\) and \(y\), you generate the integers \(a, b,\) and \(c\) that form a Pythagorean triple. For instance, substituting \(x = 6\) and \(y = 5\) into the identity gives new integers that satisfy the equation.

Finding integer solutions like these can be fascinating, as they often appear in geometric problems, such as right triangle side lengths, where the solutions are not only integers but represent real-world measurements.

Need for Mathematical Proof

Mathematical proof is a critical process of establishing the truth of a proposition. It involves logical reasoning based on previously known facts or axioms. Here, the proof of the polynomial identity illustrates how reasoning through algebra can demonstrate the equation's validity for all values of \(x\) and \(y\).

By expanding and simplifying both sides of the identity separately, and then equating the results, you're showing a clear and logical process. This step-by-step proof ensures that the identity is not just a coincidence but a universally true statement.

Understanding and creating proofs is a foundational skill in mathematics, allowing you to ensure the correctness of the concepts and techniques you apply.

Using Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions to reveal underlying patterns or solutions. It's a powerful tool in proving identities and solving equations.

In this exercise, algebraic manipulation is used to expand and simplify both sides of the polynomial identity. Techniques include distributing terms, combining like terms, and factoring when necessary.

By mastering these skills, you gain the ability to tackle a wide range of mathematical problems with confidence. It's about seeing beyond the individual terms to the bigger picture, allowing for streamlined solutions and deeper understanding.