Question: A climatologist models temperature anomalies with a cubic polynomial $ g(x) $ satisfying $ g(1) = -3 $, $ g(2) = 5 $, $ g(3) = 21 $, and $ g(4) = 55 $. Find $ g(x^2 - 1) $.
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- Thus, $ g(x^2 - 2) = 3(x^2 - 2)^3 - 22(x^2 - 2)^2 + 57(x^2 - 2) - 52 $. Expanding:
- 3(x^6 - 6x^4 + 12x^2 - 8) - 22(x^4 - 4x^2 + 4) + 57x^2 - 114 - 52 = 3x^6 - 18x^4 + 36x^2 - 24 - 22x^4 + 88x^2 - 88 + 57x^2 - 166 = 3x^6 - 40x^4 + 181x^2 - 278.
- The final answer is $ \boxed{3x^6 - 40x^4 + 181x^2 - 278} $.
- Solution: Assume $ g(x) = ax^3 + bx^2 + cx + d $. Using the given values, we form:
- a + b + c + d = -3 \\
- 8a + 4b + 2c + d = 5 \\
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