This suggests that $ f(n) = n^3 $ for $ n = 3, 4, 5, 6 $. However, $ f(x) $ is a cubic polynomial, and $ x^3 $ is already a cubic polynomial. Since a cubic is uniquely determined by four points, and $ f(n) = n^3 $ satisfies all four conditions, we conclude:

["Title: Proving $ f(x) = x^3 $ Using Data Points: Why It’s the Unique Cubic Polynomial", "Meta Description: Discover why $ f(n) = n^3 $ is the only cubic polynomial satisfying $ f(3) = 27 $, $ f(4) = 64 $, $ f(5) = 125 $, and $ f(6) = 216 $. Learn how four points uniquely determine a cubic.", "---", "Unlocking Polynomial Uniqueness: Why $ f(x) = x^3 $ Fits Perfectly", "When evaluating a cubic polynomial $ f(x) $, one might wonder: “Given the values $ f(3) = 27 $, $ f(4) = 64 $, $ f(5) = 125 $, and $ f(6) = 216 $, why do we conclude $ f(x) = x^3 $?” The answer lies in the mathematical principle that a cubic polynomial is uniquely defined by four distinct points. Let’s explore why this is the case—and why $ x^3 $ emerges as the natural solution.", "---", "### The Nature of Cubic Polynomials", "A cubic polynomial has the general form:", "[\nf(x) = ax^3 + bx^2 + cx + d\n]", "This polynomial contains four coefficients ($ a, b, c, d $) that determine its complete shape. In general, a cubic is completely specified by four independent condition points. Since $ n^3 $ yields exactly $ (3)^3 = 27 $, $ 4^3 = 64 $, $ 5^3 = 125 $, $ 6^3 = 216 $, it naturally fits the data: $ f(n) = n^3 $ gives the correct outputs at $ n = 3, 4, 5, 6 $.", "---", "### Why Only One Cubic Passes Through These Four Points", "Suppose there were two different cubic polynomials $ f(x) $ and $ g(x) $ that both passed through $ (3,27), (4,64), (5,125), (6,216) $. Then, the polynomial:", "[\nh(x) = f(x) - g(x)\n]", "is also a cubic or lower-degree polynomial, and it satisfies:", "[\nh(3) = h(4) = h(5) = h(6) = 0\n]", "That is, $ h(x) $ has four distinct roots: 3, 4, 5, and 6. But a non-zero polynomial of degree at most 3 cannot have four roots. Therefore, $ h(x) $ must be the zero polynomial, meaning $ f(x) = g(x) $. Hence, $ x^3 $ is the unique cubic polynomial consistent with these four values.", "---", "### Conclusion: $ f(n) = n^3 $ Is the Natural Fit", "Given $ f(3) = 27 $, $ f(4) = 64 $, $ f(5) = 125 $, $ f(6) = 216 $, the cubic polynomial $ f(x) = x^3 $ is not just a match—it’s the only possible cubic that meets these criteria. This demonstrates a foundational concept in polynomial interpolation: four distinct points uniquely determine a cubic polynomial, and $ n^3 $ fits perfectly.", "Understanding this principle strengthens your ability to work with mathematical models, data fitting, and functional relationships across science, engineering, and computer science.", "---", "Keywords: cubic polynomial, polynomial interpolation, $ f(n) = n^3 $, unique polynomial fit, $ n^3 $ data points, distinctive polynomial, four points define cubic, $ f(3) = 27 $, $ f(4) = 64 $", "---", "Ready to explore more? Understand how polynomials model real-world growth or solve complex data problems with a cubic foundation."]