Smallest integer making polynomial positive

Let $P(n) = n^3 - 11n^2 + 32n - 28$. Testing small integer roots: $P(2) = 8 - 44 + 64 - 28 = 0$. So $(n-2)$ is a factor.

Factoring $P(n)$: $P(n) = (n-2)(n^2 - 9n + 14) = (n-2)(n-2)(n-7) = (n-2)^2(n-7)$

We need $(n-2)^2(n-7) > 0$. Since $(n-2)^2 > 0$ for all integers $n \neq 2$, the inequality holds when: $n - 7 > 0 \implies n > 7$

The smallest integer $n > 7$ is 8.