Let P(x) be a polynomial with integer coefficients of degree d > 0. Let n be the number of distinct integer roots to P(x) = 1 or -1. Prove that n <= d + 2.
Solution
Suppose that A(x) and B(x) are two polynomials with integer coefficients which are identical except for their constant terms, which differ by 2. Suppose A(r) = 0, and B(s) =0 with r and s integers. Then subtracting we get 2 plus a sum of terms a(ri - si). Each of these terms is divisible by (r - s), so 2 must be divisible by (r - s). Hence r and s differ by 0, 1 or 2.
Now let r be the smallest root of P(x) = 1 and P(x) = -1. The polynomial with r as a root can have at most d distinct roots and hence at most d distinct integer roots. If s is a root of the other equation then s must differ from r by 0, 1, or 2. But s >= r, so s = r, r+1 or r+2. Hence the other equation adds at most 2 distinct integer roots.
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