Find the unique polynomial P(x) of degree 2 or less such that P(1)=1,P(3)=27,P(4)=64 using each of the following methods : (i) Lagrange interpolation formula:

To find the unique polynomial \( P(x) \) of degree 2 or less that satisfies the given conditions using the Lagrange interpolation formula, we can follow these steps:1. **Identify the given points:** 2. **Lagrange interpolation formula:** The Lagrange interpolation formula for a polynomial \( P(x) \) of degree 2 passing through points \((x_0, y_0)\), \((x_1, y_1)\), and \((x_2, y_2)\) is given by: where \( L_i(x) \) are the Lagrange basis polynomials defined as: 3. **Calculate the Lagrange basis polynomials:** 4. **Form the polynomial \( P(x) \):** Substitute the expressions for \( L_0(x) \), \( L_1(x) \), and \( L_2(x) \): 5. **Simplify the polynomial:** Factor out common terms: Simplify the expression inside the brackets: Combine like terms: Simplify further: Combine like terms: