MHB Why x2 +1 and x2 -1 are Not/Are Difference of Squares

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The expression x² + 1 is not a difference of squares because it cannot be factored into the form a² - b², which requires a real number b. Instead, it can be expressed as x² - (-1), leading to complex factors (x - i)(x + i). In contrast, x² - 1 is a difference of squares, as it can be factored into (x - 1)(x + 1) using real numbers. The distinction lies in the nature of the terms involved; x² + 1 involves imaginary numbers, while x² - 1 remains within the realm of real numbers. Understanding these differences is crucial for proper factorization in algebra.
Abdullah Qureshi
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Explain why x2 +1 is not a difference of squares and x2 -1 is
 
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Oh, but it is ...

$x^2+1 = x^2 - (-1) = (x - i)(x + i)$
 

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