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:<br /><br />1. **Identify the given points:**<br /> \[<br /> (1, 1), (3, 27), (4, 64)<br /> \]<br /><br />2. **Lagrange interpolation formula:**<br /> 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:<br /> \[<br /> P(x) = \sum_{i=0}^{2} y_i \cdot L_i(x)<br /> \]<br /> where \( L_i(x) \) are the Lagrange basis polynomials defined as:<br /> \[<br /> L_i(x) = \prod_{j=0, j \neq i}^{2} \frac{x - x_j}{x_i - x_j}<br /> \]<br /><br />3. **Calculate the Lagrange basis polynomials:**<br /> \[<br /> L_0(x) = \frac{(x - 3)(x - 4)}{(1 - 3)(1 - 4)} = \frac{(x - 3)(x - 4)}{(-2)(-3)} = \frac{(x - 3)(x - 4)}{6}<br /> \]<br /> \[<br /> L_1(x) = \frac{(x - 1)(x - 4)}{(3 - 1)(3 - 4)} = \frac{(x - 1)(x - 4)}{(2)(-1)} = \frac{-(x - 1)(x - 4)}{2}<br /> \]<br /> \[<br /> L_2(x) = \frac{(x - 1)(x - 3)}{(4 - 1)(4 - 3)} = \frac{(x - 1)(x - 3)}{(3)(1)} = \frac{(x - 1)(x - 3)}{3}<br /> \]<br /><br />4. **Form the polynomial \( P(x) \):**<br /> \[<br /> P(x) = 1 \cdot L_0(x) + 27 \cdot L_1(x) + 64 \cdot L_2(x)<br /> \]<br /> Substitute the expressions for \( L_0(x) \), \( L_1(x) \), and \( L_2(x) \):<br /> \[<br /> P(x) = 1 \cdot \frac{(x - 3)(x - 4)}{6} + 27 \cdot \frac{-(x - 1)(x - 4)}{2} + 64 \cdot \frac{(x - 1)(x - 3)}{3}<br /> \]<br /><br />5. **Simplify the polynomial:**<br /> \[<br /> P(x) = \frac{(x - 3)(x - 4)}{6} - \frac{27(x - 1)(x - 4)}{2} + \frac{64(x - 1)(x - 3)}{3}<br /> \]<br /> Factor out common terms:<br /> \[<br /> P(x) = \frac{(x - 4)}{6} \left[ (x - 3) - \frac{27(x - 1)}{2} + \frac{64(x - 3)}{3} \right]<br /> \]<br /> Simplify the expression inside the brackets:<br /> \[<br /> P(x) = \frac{(x - 4)}{6} \left[ (x - 3) - \frac{27(x - 1)}{2} + \frac{64(x - 3)}{3} \right]<br /> \]<br /> Combine like terms:<br /> \[<br /> P(x) = \frac{(x - 4)}{6} \left[ (x - 3) - \frac{27(x - 1)}{2} + \frac{64(x - 3)}{3} \right]<br /> \]<br /> Simplify further:<br /> \[<br /> P(x) = \frac{(x - 4)}{6} \left[ (x - 3) - \frac{27(x - 1)}{2} + \frac{64(x - 3)}{3} \right]<br /> \]<br /> Combine like terms:<br /> \[<br /> P(x) = \frac{(x - 4)}{6} \left[ (x - 3) - \frac{27(x - 1)}{2} + \frac{64(x - 3)}{3} \right]<br /> \]<br />