A nice problem on Polynomial coefficients

Question:

$P(x)$ be a polynomial with non-negative integer coefficients such that $P(0)=33$ , $P(1)=40$ and $P(9)=60000$.

Find $P(2)$.

my attempt:

Suppose $P(x) = a_nx^n + ... + a_1x + a_0$.

Now, $P(0) = 33$ implies $a_0 = 33$.

$P(1)=40\implies $sum of coeff. other than consant term $=7$

also,

$P(9)-P(0)=3.(2^5\times5^4-11)=3^2a_{1}+3^4.a_{2}+..........+a_n3^{2n}$ ,then

dividing both sides by $3^3$ we can easily infer $a_{1}=3$.

after solving it further i ended up with

$a_{2}+3a_{3}+.......a_{n}3^{2n-3}=740$

but i've to find $P(2)$ .

There are few more coefficients remaining other than $a_{1}$ whose sum is 4-and i think casework can give solution like assuming consecutive 4 coefficients $1,1,1,1$; $1,2,1$ etc. but i don't want to go that way.Any elegant solution will be appreciated .thank you

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