Roots of a polynomial are positive integers

Question

Given $a_n$ is an integer with $a_{10} = 11, a_9 = -143$, determine the number of polynomial in the form of

$$P(x) =\sum_{i=0}^{10} a_nx^n$$

such that the zeros of $P(x)$ are all positive integers.

I have never encountered a problem like this, all I know is that $a_{10}= 11$ therefore there is a factor $11x- n$ and $n$ is divisible by $11$, I don't know how to use the $a_9$.

Answer 1

Hint: Let $x_1,...,x_{10}$ be all roots of $P(x)$. Then by Vieta formulas we have

$$ x_1+...+x_{10} = -{a_{9}\over a_{10}} = 13$$

Since $x_1,...,x_{10}$ are all positive integers you can do the stars and bars method now...

Answer 2

by Vieta formula we get $$\sum_{i=1}^{10} x_i= -\frac{a_9}{a_{10}} =-\frac{-143}{11}=13$$

For any pair of positive integers n and k, the number of distinct k-tuples of positive integers whose sum is n is given by the binomial coefficient $${n - 1\choose k-1}={12\choose 9} $$.