In a certain polynomial, all the coefficients are integers, and the constant coefficient is $100000$. All the roots are integers, and distinct. Find the largest possible number of integer roots.
We recognize that there is a polynomial with only three terms and three roots: $$(x-a)(x-b)(x-c) = x^3 - (a+b+c)x^2 + (ab + bc + ac)x - abc$$ for three integers $a, b, c$.
We note that $$(x+15)(x+10)(x-8) = x^3 + 17x^2 -152x -1200$$ is such a polynomial, as all the coefficients are integers, the roots are distinct integers, and the constant coefficient is $100000$.
Since there is no limit on the degree of the polynomial, we can multiply this polynomial with others of this form to get even more integers roots. We can introduce a term $(x+1)$ on one side and a term $(x-10)$ on the other side. Here's one way of doing this: $$(x+1)(x+15)(x+10)(x-8)(x-10) = (x^2-99)(x^3 + 17x^2 -152x -1200)$$ We see that this polynomial has $5$ roots, $3$ from the original polynomial $(x+15)(x+10)(x-8)$ and $2$ from $(x^2-99)$.
We can multiply by $(x+1)$ on one side and $(x-10)$ on the other side. Here's one way: \begin{align*} &\text{LHS: } (x^2-98)(x+1)(x+15)(x+10)(x-8)(x-10) \\ &\text{RHS: } (x-10)(x^2-100)x^3 + (x-10)(2x(x^2-100)x + (x^2-100)(17x^2 -152x -1200)) \\ &= (x-10)(x^6-297x^4-76520x-1000000) \\ &=(x^7-10x^6-297x^5+3000x^4-76520x^3-997520x^2-10000000x+100000000) \end{align*} Notice that $\boxed{7}$ roots, namely $(-1,\,-15,\,-10,\,8,\,10,\,98,\,10)$ were introduced on the LHS, so the polynomial has $\boxed{7}$ integer roots.
To find the largest possible number of integer roots, we need to consider the factors of the constant coefficient, which is $100000$.
The prime factorization of $100000$ is $2^5 \cdot 5^5$.
For a polynomial with integer coefficients, the product of the roots is equal to the constant coefficient divided by the leading coefficient.
In this case, the leading coefficient is $1$, so the product of the roots will be equal to $100000$.
To find the largest possible number of integer roots, we consider the possible combinations of $2$'s and $5$'s in the product of the roots.
Since the roots must be distinct, we can have at most $5$ roots of $2$ and at most $5$ roots of $5$.
This means that the largest possible number of integer roots for the polynomial is $\boxed{10}$ (with $5$ roots of $2$ and $5$ roots of $5$).