- Find a rational polynomial such that $$P(n)=1\cdot 2+ 2\cdot 3+\cdots + n\cdot(n+1).$$ for all positive integers $n$ (edited). Does there exist an integer polynomial of this form?
Ive found that $P(X)=2\binom{n+2}{3}$ is a rational polynomial satisfying the conditions (proof by induction). I'm not sure how to approach the second part, however, probably something about interpolation...
- Prove that the polynomial $$x^{101}+101x^{100}+102$$ is irreducible over the integers (edited).
First, we can't use Perron's Criterion since $101<1+0+0+\cdots +102$. I'm not sure how to derive a contradiction either, with the little extra information given.
Help would be appreciated and a full solution rather than hints would be preferred.
For the first part, you are right that if $$P(n):=2\binom {n+2}3=\frac13n(n+1)(n+2)=\frac13n^3+n^2+\frac23n,$$ then we will hasve $P(n)=1\cdot 2+2\cdot 3+\cdots+n\cdot(n+1)$. (Note that the top of the binomial coefficient is $n+2$, not $n$.) As hinted by Gerry Myerson in the comments, if two polynomials agree at infinitely many points, then they must be the same polynomial, thus there is no integer polynomial fulfilling the conditions as $P$ has nonintegral coefficients.