---
title: "Polynomial Interpolation and Its Error Term"
description: "Fit the unique polynomial through given points, build it in Lagrange and Newton form, and bound how far it strays between the points."
canonical: https://lightmysky.com/learn/mathematics/polynomial-interpolation-and-its-error-term-mt_NSnzY9Xodd
source: https://lightmysky.com/learn/mathematics/polynomial-interpolation-and-its-error-term-mt_NSnzY9Xodd.md
retrieved: 2026-09-12
---

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# Polynomial Interpolation and Its Error Term

Fit the unique polynomial through given points, build it in Lagrange and Newton form, and bound how far it strays between the points.

Subject: Mathematics · Area: Calculus & Analysis · Ages 20 to 21
Page: https://lightmysky.com/learn/mathematics/polynomial-interpolation-and-its-error-term-mt_NSnzY9Xodd

## Ready when they can

- Construct an interpolating polynomial in both Lagrange and divided-difference form
- State the error term and use it to bound the interpolation error on an interval
- Show with a worked example why more equally spaced points can make the fit worse

## Lesson: One polynomial through the points

Give n plus 1 points with distinct x values and exactly one polynomial of degree n threads them all. A line needs two points and a quadratic needs three. This is why interpolation always works: few points pin down one low degree curve. But matching the data is not matching the function between the data. High degree interpolants can swing wildly between points while hitting every value.

Lagrange wrote the interpolant directly. For each data point build a basis piece that equals 1 at its own node and 0 at every other node. Weight each piece by its data value and add them up. Divided differences do the same job in Newton form, which is easier to extend: each new point adds one correction term built from the previous ones. Both forms produce the same unique polynomial.

The error formula mirrors the Taylor remainder. Taylor piles all the data at one point by matching value and derivatives there. Interpolation spreads the matching across the interval instead. The error is a derivative at an interior point times the product of distances to the nodes, divided by a factorial. That product term is small near the nodes and large far from them.

**Tip.** More equally spaced points can make the fit worse. Runge showed this with 1 over 1 plus 25 x squared: higher degrees oscillate violently near the ends. The fix is to cluster points near the ends, as Chebyshev points do. Where you sample matters as much as how much you sample.

**Recap.** Few points fix one polynomial, Lagrange and Newton just write it differently, and the error grows with distance from the nodes unless you cluster points at the ends.

## Practice

17 questions on this page, each with its working shown.

## Needs first

- [The Mean Value Theorem](https://lightmysky.com/learn/mathematics/the-mean-value-theorem-mt_09iApYVelx)
- [Taylor and Maclaurin Series](https://lightmysky.com/learn/mathematics/taylor-and-maclaurin-series-mt_SgI9Pn9RIO)

## Opens up

- [Numerical Quadrature: Newton-Cotes and Adaptive Rules](https://lightmysky.com/learn/mathematics/numerical-quadrature-newton-cotes-and-adaptive-rules-mt_fT5kRCUiD5)