This is a Lagrange interpolation polynomial. The general form is:

$P(x) = \sum_{i=0}^n y_i \prod_{j=0, j \neq i}^n \frac{x - x_j}{x_i - x_j}$

where $x_i$ and $y_i$ are the data points. In your case, $x_i$ are $0.00, 0.30, 0.78, 1.57, 2.09$ and $y_i$ are $3.51, 3.97, 4.19, 3.97, 3.57$.

I will expand and simplify the polynomial $P_1(x)$. sympy import symbols, expand, prod

Define variables

x = symbols('x') points = [0.00, 0.30, 0.78, 1.57, 2.09] y_values = [3.51, 3.97, 4.19, 3.97, 3.57]

Calculate Lagrange polynomial

lagrange_poly = 0 n = len(points)

for i in range(n): numerator = prod([(x - points[j]) for j in range(n) if j != i]) denominator = prod([(points[i] - points[j]) for j in range(n) if j != i]) lagrange_poly += y_values[i] * (numerator / denominator)

Simplify the polynomial

simplified_poly = expand(lagrange_poly) simplified_polyThe polynomial $P_1(x)$ simplifies to:

$P_1(x) = -0.1883x^4 + 1.0072x^3 - 2.2904x^2 + 2.1349x + 3.51$

This is the fourth-degree polynomial derived from the Lagrange interpolation method.

Would you like more details about the process or calculations? Here are some questions to deepen your understanding:

1. What is the significance of Lagrange interpolation in approximating functions?
2. How do the coefficients of $x$ relate to the behavior of the polynomial?
3. Can you use other interpolation methods, such as Newton's method, for this data set?
4. How does the degree of the polynomial affect its accuracy for interpolation?
5. How would the polynomial change if an additional data point were added?

Tip: When working with high-degree polynomials, ensure the data points are accurate, as small errors can lead to significant oscillations (Runge's phenomenon).