In a certain polynomial, all the coefficients are integers, and the constant coefficient is 42.  All the roots are integers, and distinct.  Find the largest possible number of integer roots.
By the Integer Root Theorem, any integer root must be a factor of 42.  The prime factorization of 42 is $2 \cdot 3 \cdot 7.$  Furthermore, the product of the roots is $(-1)^n \cdot \frac{42}{a_0},$ where $n$ is the degree of the polynomial, and $a_0$ is the leading coefficient.

To maximize the number of integer roots, which must be distinct, we can take the integer roots to be 2, 3, 7, 1, and $-1.$  This gives us a maximum of $\boxed{5}$ integer roots.