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Express x³−5x²+x+2 in Legendre polynomials

Short answer

Express powers using P_0=1, P_1=x, P_2=(3x²-1)/2, P_3=(5x³-3x)/2. Substitution yields f(x) = (2/5)P_3(x) - (10/3)P_2(x) + (8/5)P_1(x) + (1/3)P_0(x).

Express f(x) = x³ − 5x² + x + 2 in terms of Legendre's polynomial.

Use the Legendre polynomials P₀ = 1, P₁ = x, P₂ = (3x² − 1)/2, P₃ = (5x³ − 3x)/2.

Express powers of x in Legendre polynomials

x³ = (2P₃ + 3P₁)/5      because P₃ = (5x³ − 3x)/2  →  x³ = (2P₃ + 3x)/5
x² = (2P₂ + P₀)/3       because P₂ = (3x² − 1)/2   →  x² = (2P₂ + 1)/3
x  = P₁
1  = P₀

Substitute

f(x) = x³ − 5x² + x + 2

= (2/5)P₃ + (3/5)P₁ − (5/3)(2P₂ + P₀) + P₁ + 2P₀
= (2/5)P₃ − (10/3)P₂ + (3/5 + 1)P₁ + (2 − 5/3)P₀
= (2/5)P₃ − (10/3)P₂ + (8/5)P₁ + (1/3)P₀

Answer: x³ − 5x² + x + 2 = (2/5)P₃(x) − (10/3)P₂(x) + (8/5)P₁(x) + (1/3)P₀(x).

Check at x = 1 (all Pₙ(1) = 1): 2/5 − 10/3 + 8/5 + 1/3 = 2 − 3 = −1, and f(1) = 1 − 5 + 1 + 2 = −1 ✓.

✓ Answer checked by Claude Opus 5.5 (high effort)

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