Let-n-N-Find-the-number-of-polynomials-p-x-with-coefficients-in-0-1-2-3-such-that-p-2-n-

Question Number 117857 by bemath last updated on 14/Oct/20

Let n \in N . Find the number of

polynomials p (x) with coefficients

in {0, 1, 2, 3} such that p (2) = n

Answered by mindispower last updated on 14/Oct/20

l e t p (x) = \sum_{k ⩽ n} a_{k} x^{k}

\forall j \in [1, n - 1] a_{j} \in A = {0, 1, 2, 3}, a_{n} \in A - {0}

p (2) = n \Rightarrow 2^{n} a_{n} + 2^{n - 1} a_{n - 1} + \dots . . + a_{0} = n

2^{n} ⩾ n, \forall n \in N \Rightarrow p (2) ⩾ 2^{n} ⩾ n \Rightarrow a_{n} = 0 a b u r d

\Rightarrow p (x) = a_{0} = n, \forall n \exists! P s u c h p (2) = n