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the resulting polynomial so that .

The Lagrange interpolating polynomial for those nodes through the corresponding values is the linear combination:

Each basis polynomial has degree , so the sum has degree , and it interpolates the data because .

The interpolating polynomial is unique. Proof: assume some polynomial of degree interpolates the data. Then the difference is zero at distinct nodes $\backslash \{x\_0,\; x\_1,\; \backslash ldots,\; x\_k\backslash /math>.\}\}\; But\; the\; only\; polynomial\; of\; degree\; with\; more\; than\; roots\; is\; the\; constant\; zero\; function,\; so\; ,\; or\; .$

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