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Why is it not possible to factor a sum of squares into real numbers

User Alpav
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Answer: Factoring a sum of squares into real numbers is not always possible because the sum of squares does not always have a factorization into real numbers. This is closely related to the fact that the square root of a sum of squares is not always a rational number.

Step-by-step explanation: consider the Pythagorean theorem: a^2 + b^2 = c^2, where a, b, and c are real numbers, and c is the hypotenuse of a right triangle. In this case, a sum of squares is equal to another square, so there is a factorization: a^2 + b^2 = (c^2)^2 = c^4. However, this is a special case where the sum of squares does factor into real numbers because it's based on the Pythagorean theorem.

In more general cases, if you have a sum of squares like x^2 + y^2, you cannot always find real numbers a and b such that x^2 + y^2 = (a^2)(b^2). This is because the square root of x^2 + y^2 is √(x^2 + y^2), which is not always a rational number. It can be irrational, meaning it cannot be expressed as a fraction of two integers, and therefore, it cannot be factored into the product of two real numbers.

For example, consider x = 1 and y = 1. In this case, x^2 + y^2 = 1^2 + 1^2 = 2. The square root of 2 (√2) is an irrational number and cannot be expressed as a product of two rational (real) numbers.

In summary, while some specific cases like the Pythagorean theorem allow you to factor a sum of squares into real numbers, it's not generally possible because the square root of a sum of squares is not always a rational number, and therefore, the factorization into real numbers may not exist.

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User John Ilacqua
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