Exercise 1.4.14* (Integer-valued polynomials)

Question

Suppose that $P(x)$ is a polynomial written in the form (1.16). Show that if the $c_k$ are integers, then $P(x)$ is an integer-valued polynomial.

Richard Ganaye · Accepted Answer

ewcommand{\Z}{\mathbb{Z}}

ewcommand{\N}{\mathbb{N}}

ewcommand{\R}{\mathbb{R}}

\begin{proof} By theorem 1.21, the polynomial $P_k(x) = \binom{x}{k}$ is an integer-valued polynomial, i.e.  
$$ n \in \Z \Rightarrow P_k(n) \in \Z.$$
Let  $P(x) = \sum_{k=0}^n c_k \binom{x}{k}$. If the $c_k$ are integers, then
$$n \in \Z \Rightarrow P(n) = \sum c_k P_k(n) \in \Z,$$
therefore $P$ is an integer-valued polynomial.
\end{proof}

Exercise 1.4.14* (Integer-valued polynomials)

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Exercise 1.4.14* (Integer-valued polynomials)

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