query: If $P(x)$ is any polynomial of degree less than $n$, show that $\sum_{j=0}^n (-1)^j\binom{n}{j}P(j)=0$.
positive_passages.docid: 151384
positive_passages.text: Summation of series involving binomial coefficients and polynomial of degree at most n-1
negative_passages.docid: 18046
negative_passages.text: Prove the following equality: $\sum_{k=0}^n\binom {n-k }{k} = F_n$
negative_passages.docid: 69498
negative_passages.text: What is $\sum_{r=0}^n \frac{(-1)^r}{\binom{n}{r}}$?
negative_passages.docid: 5049
negative_passages.text: Proving $\sum_{k=0}^n(-1)^k\binom nk=0$
negative_passages.docid: 70128
negative_passages.text: Prove that $(fg)^{(n)} = \sum_{k=0}^n \binom{n}{k}f^{(k)}(x)g^{(n-k)}(x)$
negative_passages.docid: 53669
negative_passages.text: Prove that the polynomial $(x-1)(x-2)\cdots(x-n) + 1$, $ n\ge1 $, $ n\ne4 $ is irreducible over $\mathbb Z$
negative_passages.docid: 7199
negative_passages.text: Suppose that $P(x)$ is a polynomial of degree $n$ such that $P(k)=\dfrac{k}{k+1}$ for $k=0,1,\ldots,n$. Find the value of $P(n+1)$
negative_passages.docid: 1962
negative_passages.text: Polynomial $P(a)=b,P(b)=c,P(c)=a$
negative_passages.docid: 146762
negative_passages.text: Closed form for $\sum_{k=0}^{n} \binom{n}{k}\frac{(-1)^k}{(k+1)^2}$
negative_passages.docid: 96116
negative_passages.text: Show $P(z) $ is a polynomial of degree $n-1$ interpolating an Analytic function.
negative_passages.docid: 56828
negative_passages.text: A combinatorial proof that the alternating sum of binomial coefficients is zero
negative_passages.docid: 19581
negative_passages.text: $\sum_{k=0}^n (-1)^k \binom{n}{k}^2$ and $\sum_{k=0}^n k \binom{n}{k}^2$
negative_passages.docid: 40014
negative_passages.text: A polynomial that is zero on an open set
negative_passages.docid: 94406
negative_passages.text: Degree of minimum polynomial at most $n$ without Cayley-Hamilton?
negative_passages.docid: 155444
negative_passages.text: Show that entire function $f$ is a polynomial of degree at most $n$
negative_passages.docid: 116461
negative_passages.text: Prove the identity $\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}=0$ using a combinatorial interpretation of the positive and negative terms
negative_passages.docid: 92704
negative_passages.text: Prove that $\sum_{k=1}^{n}k\binom{n}{k}=n\cdot2^{n-1}$