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  "body": "\n## Summary\n\nTake $n + 1$ points in the plane with distinct first coordinates.\nTheir Lagrange polynomial is the unique [[polynomial function|1G0B73WPX8]] of degree at most $n$ whose graph passes through all of them.\nIt is written as a sum of basis polynomials, each equal to $1$ at one node and $0$ at the others.\nIt gives an explicit solution of [[polynomial interpolation|SS1YMS2NPC]] and is the starting point of numerical differentiation and quadrature rules.\nThe [[definition|TEQNAME853]] gives the formula.\n\n## Explanations\n\n### L1 human\n\nThrough two points there is one line, and through three points with different $x$-values there is one parabola or line.\nThe Lagrange formula writes this curve down directly.\nFor each point it builds a small polynomial that is $1$ at that point and $0$ at all the other points.\nThen it adds these pieces, each multiplied by the height it must reach.\nFor the points $(0, 1)$, $(1, 3)$, $(2, 7)$ the result is $x^2 + x + 1$.\n",
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