Math 2.0

Approximations and expansions Definition

Polynomial interpolation

Builds on FieldPolynomial ringDegree of a polynomialA polynomial over a field has at most as many roots as its degree

Definition

Let KK be a field, let n≥0n \ge 0, let x0,…,xn∈Kx_0, \dots, x_n \in K be pairwise distinct, and let y0,…,yn∈Ky_0, \dots, y_n \in K. An interpolation polynomial of the data (xi,yi)(x_i, y_i) is a polynomial p∈K[x]p \in K[x] such that:

  1. p=0p = 0 or deg⁡p≤n\deg p \le n;
  2. p(xi)=yip(x_i) = y_i for i=0,…,ni = 0, \dots, n.

Remarks

There is exactly one interpolation polynomial. Uniqueness: the difference of two of them has degree at most nn and n+1n + 1 roots, so it is 00 by the bound on the number of roots. Existence: the Lagrange formula gives one. Equivalently, the coefficients solve a linear system whose matrix has the nonzero Vandermonde determinant. Example: for (0,1)(0, 1), (1,3)(1, 3), (2,7)(2, 7) the interpolation polynomial is x2+x+1x^2 + x + 1. Non-example: x2+x+1+x(x−1)(x−2)x^2 + x + 1 + x(x - 1)(x - 2) takes the same values at 0,1,20, 1, 2, but it has degree 3>23 > 2, so clause 1 fails.

Reading path

To understand this page from the ground up, read these 20 pages in order. The foundations come first.

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Group, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionCommutative ring, definition, no evidence levelCommutative ringdefinitionPolynomial ring, definition, no evidence levelPolynomial ringdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionZero divisor, definition, no evidence levelZero divisordefinitionDegree of a polynomial, definition, no evidence levelDegree of a polynomialdefinitionField, definition, no evidence levelFielddefinitionPolynomial evaluation, definition, no evidence levelPolynomial evaluationdefinitionIntegral domain, definition, no evidence levelIntegral domaindefinitionLeading coefficient, definition, no evidence levelLeading coefficientdefinitionRoot of a polynomial, definition, no evidence levelRoot of a polynomialdefinitionMonic polynomial, definition, no evidence levelMonic polynomialdefinitionMultiplicity of a root, definition, no evidence levelMultiplicity of a rootdefinitionDegree of a product, statement, no evidence levelDegree of a productstatementDivision algorithm for polynomials, statement, no evidence levelDivision algorithm fo…statementRemainder theorem, statement, no evidence levelRemainder theoremstatementFactor theorem, statement, no evidence levelFactor theoremstatementA polynomial over a field has at most as many roots as its degree, statement, no evidence levelA polynomial over a f…statementPolynomial interpolation, definition, no evidence levelPolynomial interpolat…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Group concept
  2. Ring definition
  3. Commutative ring definition
  4. Polynomial ring definition
  5. Unit of a ring definition
  6. Zero divisor definition
  7. Degree of a polynomial definition
  8. Field definition
  9. Polynomial evaluation definition
  10. Integral domain definition
  11. Leading coefficient definition
  12. Root of a polynomial definition
  13. Monic polynomial definition
  14. Multiplicity of a root definition
  15. Degree of a product statement
  16. Division algorithm for polynomials statement
  17. Remainder theorem statement
  18. Factor theorem statement
  19. A polynomial over a field has at most as many roots as its degree statement
  20. Polynomial interpolation definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Polynomial_interpolation
About this page
TagEKDSDN89FR
URIm2://EKDSDN89FR
Kinddefinition
WritingW0 stub
MSC41A05, 65D05
DefinesPolynomial interpolation
UsesField, Polynomial ring, Degree of a polynomial, A polynomial over a field has at most as many roots as its degree
Changed2 November 2026

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