Word Count : 414

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The given data points are: $$ (x_0, y_0) = (1, -32), \quad (x_1, y_1) = (4, 8), \quad (x_2, y_2) = (5, 52), \quad (x_3, y_3) = (7, 167) $$ The Lagrange interpolation polynomial is: $$ P(x) = \sum_{i=0}^{3} y_i \, L_i(x) $$ where $$ L_i(x) = \prod_{j=0, j\neq i}^{3} \frac{x - x_j}{x_i - x_j} $$ Step 1: Compute $L_0(x)$ $$ L_0(x) = \frac{(x-4)(x-5)(x-7)}{(1-4)(1-5)(1-7)} = \frac{(x-4)(x-5)(x-7)}{(-3)(-4)(-6)} = \frac{(x-4)(x-5)(x-7)}{-72} $$ Step 2: Compute $L_1(x)$ $$ L_1(x) = \frac{(x-1)(x-5)(x-7)}{(4-1)(4-5)(4-7)} = \frac{(x-1)(x-5)(x-7)}{3 \cdot (-1) \cdot (-3)} = \frac{(x-1)(x-5)(x-7)}{9} $$ Step 3: Compute $L_2(x)$ $$ L_2(x) = \frac{(x-1)(x-4)(x-7)}{(5-1)(5-4)(5-7)} = \frac{(x-1)(x-4)(x-7)}{4 \cdot 1 \cdot (-2)} = \frac{(x-1)(x-4)(x-7)}{-8} $$ _______ ____ _________ ____ _______ _______ _______ ______ ______ ________ ___ ______.
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