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We sometimes express polynomials as $p(x)=\sum_{k=0}^na_kx^k$. If I wanted to see the value of $p$ at $0$, naively, I would just substitute $0$ everywhere I see $x$. So

$p(0)=\sum_{k=0}^na_k\cdot 0^k$

$=a_0\cdot 0^0+a_1\cdot 0^1+\dots+a_n\cdot 0^n$

but... this has a $0^0$. What have I done wrong?

fdgfdgfdgfdg
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  • $variable^0 = 1$ always. $0^{variable} = 0$ always. The confusion only arises when both are constant and we don't know what rule to apply. That's not the case here. We are allow to say $x^0:x=0$ evaluates to 2. – fleablood Aug 15 '16 at 02:50

1 Answers1

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There is nothing wrong. $0^0 = 1$, which makes sense since the constant term of any polynomial does not depend on the input $x$.

IntegrateThis
  • 3,778
  • yes, but http://math.stackexchange.com/questions/11150/zero-to-the-zero-power-is-00-1. Why should something as basic as substituting equals for equals depend on the meaning of $0^0$, which is apparently something people can't even agree on? – fdgfdgfdgfdg Aug 15 '16 at 02:08
  • For an application this simple it shouldn't really matter. Besdies the alternative would be that in this case $0^0 = 0$ therefore every polynomial at 0 is the 0 function?? – IntegrateThis Aug 15 '16 at 02:13
  • I really think there's something improper about this. You say it shouldn't matter in this case, but.... then when does it matter? I think the answer here is that we mentally rewrite the function as $\sum_{k=1}^na_kx^k+a_0$ (this is the proper form). – fdgfdgfdgfdg Aug 15 '16 at 02:22