Freshman's dream

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The freshman's dream is a name sometimes given to the error $(x + y) n = x n + y n$ , which is commonly made among beginning students. When n = 2, it is easy to see why this is incorrect: $(x + y) 2$ can be correctly computed by FOILing as $x 2 + 2 x y + y 2$ . For larger values of n, the correct result is given by the binomial theorem.

The name is also sometimes given to the theorem that says that $(x + y) p = x p + y p$ if x and y are members of a field of characteristic p; in this case, the "mistake" actually gives the correct result.