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  "body": "\n## Summary\n\nLagrange interpolation is an explicit formula for the polynomial of least degree through given points.\nIt writes the [[interpolation polynomial|SS1YMS2NPC]] as a combination of basis polynomials, each equal to $1$ at one node and $0$ at the others.\nThe formula works over every field and shows at once that the interpolation polynomial exists.\nIt is used in numerical analysis, in coding theory and in secret sharing.\n\n## Explanations\n\n### L1 human\n\nSuppose you want a polynomial that takes chosen values at chosen points.\nFirst build, for each point, a polynomial that is $1$ there and $0$ at all other points.\nThen multiply each of them by the wanted value and add.\nFor the points $(0, 1)$, $(1, 3)$ and $(2, 7)$ this gives $x^2 + x + 1$.\nThe [[definition|QGQTYT6YZR]] gives the formula.\n",
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