### Instruction: Find all values of $b$ for which the equations $1988x^2 + bx + 8891 = 0$ and $8891x^2 + bx + 1988 = 0$ have a common root.

Enter all the possible values of $b,$ separated by commas.### Response: To find the values of $b$ that make the equations have a common root, I need to find the values of $x$ that satisfy both equations.
I can try to eliminate $b$ by subtracting the second equation from the first.
This gives me $(-6903)x^2 + 6903 = 0,$ which simplifies to $x^2 - 1 = 0.$
This equation has two roots: $x = \pm 1.$
So I need to find the values of $b$ that make either $x = 1$ or $x = -1$ a root of both equations.
I can plug in $x = 1$ into the first equation and get $1988 + b + 8891 = 0.$
Solving for $b,$ I get $b = -10879.$
I can check that this value of $b$ also makes $x = 1$ a root of the second equation by plugging it in and verifying that it equals zero.
Similarly, I can plug in $x = -1$ into the first equation and get $1988 - b + 8891 = 0.$
Solving for $b,$ I get $b = 10879.$
I can check that this value of $b$ also makes $x = -1$ a root of the second equation by plugging it in and verifying that it equals zero.
Therefore, the only values of $b$ that make the equations have a common root are $b = \pm 10879.$