Problem
ALG-B1-M04-P015 How Many Values Are Enough
A teacher chooses a monic polynomial of degree \(5\) with integer coefficients. He wants to name \(k\) distinct integer points and the product of the polynomial values at those points so that the polynomial is determined uniquely. Prove that the smallest possible \(k\) is \(5\).
For \(k<5\), add the product \((x-n_i)\). For \(k=5\), choose points far apart and product \(1\).
If \(k<5\), then from any suitable polynomial \(P\) we can build another one: \(Q(x)=P(x)+(x-n_1)\cdots(x-n_k)\). It remains monic of degree \(5\), but \(Q(n_i)=P(n_i)\). Thus uniqueness is impossible.
For \(k=5\), choose \(n_i=10i\) and announce that the product of values is \(1\). The polynomial \(P(x)=\prod_{i=1}^5(x-n_i)+1\) works.
If another monic integer polynomial \(Q\) also gives product \(1\), then all \(Q(n_i)=\pm1\). But \(Q(n_i)-Q(n_j)\) is divisible by \(n_i-n_j\), while \(2\) is not divisible by \(10\) or more. Hence all values \(Q(n_i)\) are equal; since there are five of them and their product is \(1\), all equal \(1\). Then \(P-Q\) has five roots and degree at most \(4\), so \(P=Q\).
A scaled-down version of a strong idea about integer values and interpolation.