MHB Polynomial $P(x)$ Satisfying $P(k)=2^k$: POTW #396 Dec 9th, 2019

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A polynomial \( P(x) \) of degree \( n \) satisfies \( P(k) = 2^k \) for integers \( k = 0, 1, 2, \ldots, n \). The goal is to determine the value of \( P(n+1) \). The solution involves using the properties of polynomials and interpolation techniques. Members Opalg and kaliprasad successfully provided correct solutions to the problem. The discussion emphasizes the importance of understanding polynomial behavior at specific integer points.
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Here is this week's POTW:

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A polynomial $P(x)$ of the $n$-th degree satisfies $P(k)=2^k$ for $k=0,\,1,\,2,\,\cdots,\,n$. Find the value of $P(n+1)$.

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Congratulations to the following members for their correct solution!(Cool)

1. Opalg
2. kaliprasad

Solution from Opalg:
If $P(x)$ has degree $n$ then the difference polynomial $P(x) - P(x-1)$ has degree $n-1$, because the coefficients of $x^n$ cancel.The second difference $(P(x) - P(x-1)) - (P(x-1) - P(x-2)) = P(x) - 2P(x-1) + P(x-2)$ has degree $n-2$, and so on.

More precisely, define a sequence of polynomials $P_k(x)$ inductively by $P_0(x) = P(x)$ and $P_{k+1}(x) = P_k(x) - P_k(x-1)$ for $k\geqslant0$. Then $P_k(x)$ is a polynomial of degree $n-k$. A proof by induction, using the fact that ${k+1\choose j} = {k\choose j} + {k\choose j-1}$, shows that $$P_k(x) = \sum_{j=0}^k (-1)^j{k\choose j}P(x-j)$$.

When $k=n$, $P_n(x)$ will be a polynomial of degree $0$, a constant. And when $k=n+1$, $P_{n+1}(x) = P_n(x) - P_n(x-1)$ will be identically zero. In particular, $P_{n+1}(n+1) = 0$. But $$P_{n+1}(n+1) = \sum_{j=0}^{n+1} (-1)^j{n+1\choose j}P(n+1-j) = P(n+1) + \sum_{j=1}^{n+1} (-1)^j{n+1\choose j}2^{n+1-j}.$$ Therefore $$P(n+1) + \sum_{j=1}^{n+1} (-1)^j{n+1\choose j}2^{n+1-j} = 0.\qquad(*)$$ On the other hand, the binomial expansion of $(x-1)^{n+1}$, with $x=2$, shows that $$2^{n+1} + \sum_{j=1}^{n+1} (-1)^j{n+1\choose j}2^{n+1-j} = (2-1)^{n+1} = 1.$$ Compare that last equation with $(*)$ to see that $P(n+1) = 2^{n+1} - 1.$
 
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