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The freshman's dream is a name sometimes given to the error (''x'' + ''y'')''n'' = ''x''''n'' + ''y''''n'', where ''n'' is a real number (usually a positive integer greater than 1). Beginning students commonly make this error in computing the power of a sum of real numbers.〔Julio R. Bastida, ''Field Extensions and Galois Theory'', Addison-Wesley Publishing Company, 1984, p.8.〕〔Fraleigh, John B., ''A First Course in Abstract Algebra'', Addison-Wesley Publishing Company, 1993, p.453, ISBN 0-201-53467-3.〕 When ''n'' = 2, it is easy to see why this is incorrect: (''x'' + ''y'')2 can be correctly computed as ''x''2 + 2''xy'' + ''y''2 using distributivity (or commonly known as the FOIL method). For larger positive integer values of ''n'', the correct result is given by the binomial theorem. The name "freshman's dream" also sometimes refers to the theorem that says that for a prime number ''p'', if ''x'' and ''y'' are members of a commutative ring of characteristic ''p'', then (''x'' + ''y'')''p'' = ''x''''p'' + ''y''''p''. In this case, the "mistake" actually gives the correct result, due to ''p'' dividing all the binomial coefficients save the first and the last. ==Examples== *, but . * does not generally equal . For example, , which does not equal 3+4=7. In this example, the error is being committed with the exponent ''n'' = . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Freshman's dream」の詳細全文を読む スポンサード リンク
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