Word Count : 350

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Rodrigue's formula for Legendre polynomials is given by: \[ P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2 - 1)^n \] Now, let’s compute \( P_3(x) \) and \( P_4(x) \). ### Step 1: Compute \( P_3(x) \) For \( n = 3 \), \[ P_3(x) = \frac{1}{2^3 3!} \frac{d^3}{dx^3} (x^2 - 1)^3 \] Expanding \( (x^2 - 1)^3 \): \[ (x^2 - 1)^3 = x^6 - 3x^4 + 3x^2 - 1 \] Now, compute the third derivative: 1. First derivative: \[ \frac{d}{dx} (x^6 - 3x^4 + 3x^2 - 1) ___ __________ ______ ________ __________.
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