TITLE: How can I get to Mars with a polynomial?
QUESTION [11 upvotes]: In order to get to Mars you must win a video game. The video game
  chooses $10$ points $(a_i,b_i)$ where $a_i$ and $b_i$ are single-digit
  integers, and places a disk with radius $1/3$ on each of the
  points. You must find a polynomial $f$ such that the graph of $f$ hits
  all $10$ discs.
However, you must choose your polynomial before seeing where the disks
  are. Find a polynomial that guarantees you a trip to mars.

The only clue I have for this problem is that for some point $x=h$ I must have a polynomial which is quite "steep" at that point and resembles the graph of a vertical line.In this way I am guaranteed that any other lattice point $(a_i,b_i)$ above it is included.
The only way I can imagine a polynomial doing that is when I have a product of positive expressions, i.e for $x=h$ I would have something like $(x-h)(x+r)(x+d)$ where $r,d$ are some large numbers.
The main conceptual difficulty I am facing here is to produce this kind of behavior for every single digit integer on the number line...

REPLY [12 votes]: We know that $(a,b)\in[0,0]\times [9,9]$ (since $a,b$ are single-digit integers. If negative numbers are considered "single-digit integers", too, then this is generalizable for those, too). So, what about $$p(x)=100(x-0)(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)(x-7)(x-8)(x-9)$$
A plot made with Wolfram Mathematica 10.0:

So if you wanna go to Mars, this is probably a good polynomial to try.