OMO (1)

Calculus Level 3

Suppose that $g$ and $h$ are polynomials of degree 10 with integer coefficients such that $g(2) < h(2)$ and

$g(x)h(x) =\sum_{k=0}^{10} \Bigg(\binom{k+11}{k}x^{20-k} - \binom{21- k }{11} x^{k-1} + \binom{21}{11}x^{k-1}\Bigg)$

holds for all nonzero real numbers $x$ . What is $g(2)=?$

2047 2051 2050 2046 2048

1 solution

Hana Wehbi
May 12, 2018

Relevant wiki: Hockey Stick Identity

Let $n=10$ . By the Hockey Stick Identity, one can say the factorization is:

$p(x)= (x^n+x^{n-1}+x^{n-2}+\dots+x+1)\Bigg(\sum_{i=0}^{n}\binom{n+i}{n}x^{n-i}\Bigg)$

The polynomial $x^{10}+x^9+\dots+1$ happens to be irreducible ( it is the $11th$ cyclotomic polynomial), and so this must be the unique factorization into two polynomials of degree $n$ . Hence, the answer is just $1+2+\dots+2^{10}=2047.$

OMO (1)

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