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x^4 + ( -2x^3 + 2x^3 ) + (3x^2 - 4x^2 + 3x^2) + (6x - 6x) + 9 = x^4 + 2x^2 + 9
Final answer: $ \boxed{x^4 + 2x^2 + 9} $Question: A cartographer is analyzing elevation data from a triangular region defined by points $ A = (1, 2), B = (4, 6), C = (7, 3) $. If the elevation at each vertex is proportional to its respective $ z $-coordinate, and the average elevation over the triangle is $ 12 $, what is the $ z $-coordinate of the centroid of triangle $ ABC $?
Solution: The centroid $ G $ of triangle $ ABC $ is the average of its vertices' coordinates:

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