\begin{tabular}{l}\hline Exercise 4.14 \\ Consider data points \( (0,2),(1,5) \) and \( (2,25) \). Find an interpolating polynomial \( p(x) \) \\ of at most degree two and estimate the values of \( p(-1) \) and \( \rho(3) \). \\ \end{tabular}

To find an interpolating polynomial \( p(x) \) of at most degree two that passes through the points \( (0,2) \), \( (1,5) \), and \( (2,25) \), we can use the Lagrange interpolation formula or construct a polynomial in the form \( p(x) = ax^2 + bx + c \). ### Step 1: Set up the polynomial Assume the polynomial is of the form: \[ p(x) = ax^2 + bx + c \] ### Step 2: Create a system of equations We will use the given points to create a system of equations: 1. For the point \( (0, 2) \): \[ p(0) = c = 2 \] 2. For the point \( (1, 5) \): \[ p(1) = a(1)^2 + b(1) + c = 5 \implies a + b + 2 = 5 \implies a + b = 3 \] 3. For the point \( (2, 25) \): \[ p(2) = a(2)^2 + b(2) + c = 25 \implies 4a + 2b + 2 = 25 \implies 4a + 2b = 23 \implies 2a + b = 11.5 \] ### Step 3: Solve the system of equations Now we have the following system of equations: 1. \( a + b = 3 \) 2. \( 2a + b = 11.5 \) We can solve this system. Let's subtract the first equation from the second: \[ (2a + b) - (a + b) = 11.5 - 3 \] \[ a = 8.5 \] Now substitute \( a = 8.5 \) back into the first equation: \[ 8.5 + b = 3 \implies b = 3 - 8.5 = -5.5 \] Now we have: - \( a = 8.5 \) - \( b = -5.5 \) - \( c = 2 \) ### Step 4: Write the polynomial Thus, the interpolating polynomial is: \[ p(x) = 8.5x^2 - 5.5x + 2 \] ### Step 5: Estimate \( p(-1) \) and \( p(3) \) Now we will calculate \( p(-1) \) and \( p(3) \). 1. **Calculate \( p(-1) \)**: \[ p(-1) = 8.5(-1)^2 - 5.5(-1) + 2 = 8.5 + 5.5 + 2 = 16 \] 2. **Calculate \( p(3) \)**: \[ p(3) = 8.5(3)^2 - 5.5(3) + 2 = 8.5(9) - 16.5 + 2 = 76.5 - 16.5 + 2 = 62 \] ### Final Results Thus, the values are: - \( p(-1) = 16 \) - \( p(3) = 62 \)