Question 100
a. Let $a_0$, $a_1$, $a_2$ be real numbers and consider the polynomial
$$
P(x) = a_0 + a_1 x + a_2 x^2.
$$
Assume that $P(-1)$, $P(0)$ and $P(1)$ are integers.
Prove that $P(n)$ is an integer for all integers $n$.
b. Let $a_0$, $a_1$, $a_2$, $a_3$ be real numbers and consider the polynomial
$$
Q(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3.
$$
Assume that there exists an integer $i$ such that $Q(i)$, $Q(i+1)$, $Q(i+2)$ and $Q(i+3)$ are integers.
Prove that $Q(n)$ is an integer for all integers $n$.
$$
P(x) = a_0 + a_1 x + a_2 x^2.
$$
Assume that $P(-1)$, $P(0)$ and $P(1)$ are integers.
Prove that $P(n)$ is an integer for all integers $n$.
b. Let $a_0$, $a_1$, $a_2$, $a_3$ be real numbers and consider the polynomial
$$
Q(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3.
$$
Assume that there exists an integer $i$ such that $Q(i)$, $Q(i+1)$, $Q(i+2)$ and $Q(i+3)$ are integers.
Prove that $Q(n)$ is an integer for all integers $n$.