# The Rational Zero Test, the Remainder Theorem, and the Mysterious Cabin

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Instructions page 1: A mysterious abandoned cabin is located in the middle of a forest as indicated by an old handwritten map. According to the map’s author (a mathematician/explorer named Doreena Shadetree, who lived around the early 1900s), if you lay a grid (with each square inch equal to a square mile) over the map, where the cabin is at (0,0), north is up and left is west, a polynomial with equation y = 4x3 + 4x2 - x - 1, will give the location of a mysterious and winding dirt road. There is a cobblestone pathway that runs west to east, represented by the x-axis of the grid on the map. Where the winding dirt road intersects the cobblestone road are hidden keys that together will open the cabin. Also, according to Doreena’s notation there is hidden treasure inside the cabin! Use the rational zero test, to determine the possible real zeros (locations where the keys may be hidden). Then, test these possible real zeros, using the remainder theorem and synthetic division, to determine which are actually real zeros! These are where the keys are! Make sure to show all work! At the bottom of this page sketch a picture of the winding road, the cobblestone road, the cabin, and the location of the keys. Also, at the bottom write how far east and west of the cabin the keys are located.

Page 2 is a similar exercise with a different polynomial

Page 3 is the answer key (work not shown)

Total Pages
3 pages
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Teaching Duration
50 minutes
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