 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ib/paper-1/18c/c/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

q_{4} (x) = \frac{1}{8} (35 x^{4} - 30 x^{2} + 3) .

A four-node Gaussian rule is exact through degree

2 \cdot 4 - 1 = 7

. Solving

q_{4} (x) = 0

first for

x^{2}

gives the four nodes

c_{k} \in ⎩ ⎨ ⎧ - \frac{15 + 2 30}{35}, - \frac{15 - 2 30}{35}, \frac{15 - 2 30}{35}, \frac{15 + 2 30}{35} ⎭ ⎬ ⎫ .

The rule is not exact for every polynomial of degree eight: its quadrature value on

q_{4}^{2}

is zero because every node is a root of

q_{4}

, whereas

\int_{- 1}^{1} q_{4} (x)^{2} d x > 0.

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