A 5-point Gaussian-Legendre quadrature rule exactly integrates all polynomials of degree up to:
A4
B5
C9
D10
An n-point Gaussian quadrature rule exactly integrates all polynomials of degree up to 2n − 1. With n = 5 points, this is 2(5) − 1 = 9. The common mistake is thinking a 5-point rule handles only degree-5 polynomials. The reason for the 2n − 1 exactness comes from orthogonal polynomial theory: any degree-(2n−1) polynomial can be written as P_n(x)·s(x) + r(x) where s and r have degree less than n; the first term integrates to zero by orthogonality, and the n-point rule captures r(x) exactly.
Question 2 Multiple Choice
Chebyshev polynomials and Legendre polynomials both form orthogonal families on [-1, 1]. What is the key difference between them?
ALegendre polynomials have more roots on [-1, 1] than Chebyshev polynomials of the same degree
BThey are orthogonal with respect to different weight functions: Legendre uses w(x) = 1, Chebyshev uses w(x) = 1/√(1−x²)
CChebyshev polynomials are not actually orthogonal — they just minimize approximation error
DLegendre polynomials are only defined for even degrees
Both families are orthogonal polynomials on [-1, 1], but the inner product differs. Legendre: ⟨f,g⟩ = ∫₋₁¹ f(x)g(x) dx (uniform weight). Chebyshev: ⟨f,g⟩ = ∫₋₁¹ f(x)g(x)/√(1−x²) dx (endpoint-upweighting). The Chebyshev weight concentrates attention near the endpoints, which is why Chebyshev polynomials have the minimax property — they minimize the maximum deviation from zero among all monic polynomials, making them optimal for polynomial approximation in the sup-norm sense.
Question 3 True / False
The n roots of the nth orthogonal polynomial P_n are the optimal node locations for an n-point Gaussian quadrature rule.
TTrue
FFalse
Answer: True
This is the central connection between orthogonal polynomials and Gaussian quadrature. The roots of P_n are called Gauss points or quadrature nodes. Using them as evaluation points, with carefully chosen quadrature weights, produces a rule that integrates all polynomials of degree up to 2n−1 exactly. No other choice of n nodes achieves this degree of exactness — the orthogonal polynomial roots are uniquely optimal.
Question 4 True / False
Gaussian quadrature with n points is generally less accurate than an n-point equally-spaced rule (like the composite trapezoidal rule) for smooth functions.
TTrue
FFalse
Answer: False
The opposite is true — Gaussian quadrature is dramatically more accurate. For smooth functions, Gaussian quadrature converges exponentially fast as n increases, not just algebraically. A 5-point Gaussian rule integrates all polynomials of degree up to 9 exactly, while a 5-point composite trapezoidal or Simpson's rule achieves only algebraic convergence. The orthogonal polynomial node placement exploits the full polynomial-exactness budget of n function evaluations.
Question 5 Short Answer
Why does a 5-point Gaussian-Legendre rule exactly integrate polynomials of degree up to 9, rather than just up to 4? What property of orthogonal polynomials makes this possible?
Think about your answer, then reveal below.
Model answer: An n-point rule has 2n free parameters (n node locations plus n weights). By choosing nodes at the roots of P_n and weights optimally, the rule is exact for all polynomials of degree up to 2n−1 — not just n−1. The key is that any degree-(2n−1) polynomial can be decomposed as P_n(x)·s(x) + r(x) where s and r have degree less than n. The first term integrates to zero by orthogonality of P_n; the second is captured exactly by the n-point rule. This doubles the polynomial exactness degree.
The orthogonality of P_n is doing crucial work: it guarantees that the 'high-degree' component P_n·s integrates to zero for free, without using any of the n function evaluations. Only the degree-(n−1) remainder r(x) needs to be handled explicitly, which n points can do exactly. This argument breaks down for degree ≥ 2n because the remainder r would have degree ≥ n and would not be exactly integrable.