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 · Let , → be a continuous function on the closed interval ,, and differentiable on the open interval (,), where < Then there exists some in (,) such that ′ = () The mean value theorem is a generalization of Rolle's theorem, which assumes () = (), so that the righthand side above is zero The mean value theorem is still valid in a slightly more general setting.

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G(x) ∈ Zx such that f(x)g(x) = x p − x mod p as polynomials (all coefficients mod p) Proof Suppose f(x) has n solutions Then n ≤ p because only p possible roots mod p (ie, deg(f) ≤ deg(x p − x)) Divide x p − x by f(x) to get x p − x = f(x)g(x) r(x), deg(r) < deg(f) = n Now note, if α is a root of f(x) mod p then. ETC f 梃X s h } X ^ o L O Z J !. Z z z m m o N 07 c o O o O 00 — z Ccn m 0 Created Date 6/28/19 AM.

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