If a polynomial $P$ with integer coefficients has three distinct integer zeros, then show that $P(n)\neq1$ for any integer $n$.
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Hint: first suppose that $P(x)$ has degree $3$. Can you factor it? – lulu May 03 '16 at 10:22
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1$q(n)(n-a)(n-b)(n-c)=1$ with $n,a,b,c$ integers. – almagest May 03 '16 at 10:28
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Sketch:
Suppose that $P(a)=P(b)=P(c)=0$ where $a,b,c$ are integers.
Suppose that there exists a $d$ such that $P(d)=1$.
Prove that $(d-a)\mid P(d)-P(a)=1$. Hint: Prove $(d-a)\mid (d^k-a^k)$ by factoring.
This gives three distinct divisors of $1$, do you detect a contradiction?
Michael Burr
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