Fundamental Thm. of Algebra
& Rational Roots Theorem

  1. The polynomial \(x^2-4x+13\) doesn’t have real roots. What exactly are its complex roots?
  2. Without the aid of technology, confirm that \(x=1\) is a root of this polynomial, and calculate what the other two roots must be. \[2x^3+x^2-10x+7\]
  3. Without the aid of technology, confirm that \(x=-2\) and \(x=5\) are each a root of this polynomial, and calculate what the other two roots must be. \[2x^4-5x^3-24x^2-7x+10\]
  4. Knowing that the four roots of this polynomial are all rational, completely factor it. \[ 12x^4 + 16x^3 - 59x^2 +6x +9 \]

Puzzle

On a remote mountaintop there’s a small village of one hundred people. Everyone in the village has either brown eyes or blue eyes, and devoutly follows the same religion that has a single peculiar feature: each villager is forbidden from knowing the color of their own eyes, or even from discussing the topic. So each villager sees the eye color of every other villager, but has no way of discovering their own. (They’ve forbidden reflective surfaces long ago.) If a villager does discover their eye color, their religion compels them to proceed to the village square at high noon the following day, and call out for all their fellow villagers to come witness as they exile themselves from the village forever.

Of the one hundred villagers, it turns out that ten of them have blue eyes and the rest have brown eyes, although they are not aware of this fact. Each villager is highly logical and religiously devout, and they all know that each villager is highly logical and devout (and they all know that they all know that each villager is highly logical and religiously devout, and …).

One day a blue-eyed mountaineer, tired and hungry, stumbles across the village asking for help. That evening at the communal village dinner she addresses the whole gathering to thank them for their hospitality. However, not knowing their customs, the mountaineer remarks in her address how unusual it is to see another blue-eyed person like myself in this remote region. What effect, if any, does this faux pas have on the village?