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  "body": "\n## Summary\n\nThe Newton polynomial is the [[interpolation polynomial|SS1YMS2NPC]] of given data written in the Newton form.\nThe basis is $1$, $(x - x_0)$, $(x - x_0)(x - x_1)$, and so on, and the coefficients are [[divided differences|N5ZHV4HWDC]].\nAdding one more data point adds one more term and leaves the earlier terms unchanged.\nThis makes the Newton form convenient for computation and for deriving error terms and finite difference formulas.\n\n## Explanations\n\n### L1 human\n\nStart with the constant through the first point.\nAdd a multiple of $(x - x_0)$ to also pass through the second point; this does not change the value at $x_0$.\nThen add a multiple of $(x - x_0)(x - x_1)$ to pass through the third point, without disturbing the first two.\nFor the points $(0, 1)$, $(1, 3)$, $(2, 7)$ this gives $1 + 2x + x(x - 1) = x^2 + x + 1$.\nThe [[definition|155K06F4JF]] gives the general formula.\n",
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