### Instruction: If $A$ and $B$ are numbers such that the polynomial $x^{2017} + Ax + B$ is divisible by $(x + 1)^2$, what is the value of $B$?### Response: I need to find out what conditions on $A$ and $B$ would make the polynomial divisible by $(x + 1)^2$.
I know that if a polynomial $p(x)$ is divisible by $(x + 1)^2$, then $p(-1) = 0$ and $p'(-1) = 0$, where $p'(x)$ is the derivative of $p(x)$.
So I can try to apply this criterion to my polynomial and see what I get.
Plugging in $x = -1$, I get $(-1)^{2017} + A(-1) + B = 0$, which simplifies to $-1 - A + B = 0$, or $B = A + 1$.
Differentiating the polynomial, I get $2017x^{2016} + A$.
Plugging in $x = -1$, I get $2017(-1)^{2016} + A = 0$, which simplifies to $2017 + A = 0$, or $A = -2017$.
Now I have both $A$ and $B$, so I can answer the question.
Since $B = A + 1$ and $A = -2017$, I get $B = -2017 + 1 = -2016$.