Verify that the polynomial p(x)=∑i=0^k di ∏j ≠i (x-rj)/(ri-rj) satisfies p(ri)=di for i=0,1, …,

Step 1:u000AUnderstand the polynomial expression given:u000A$$ p(x) u003D u005Csum_{iu003D0}^k d_i u005Cprod_{j u005Cneq i}

Verify that the polynomial p(x)=∑i=0^k di ∏j ≠i...

Key Concepts

Lagrange Basis Polynomials

Lagrange Basis Polynomials are the components of the Lagrange Interpolating Polynomial. Each basis polynomial is constructed to be equal to one at its own corresponding interpolation node and zero at all others. This property, often referred to as the Kronecker delta property, ensures that when the interpolation polynomial is evaluated at any given node, only the corresponding term contributes, yielding the correct function value.

Polynomial Interpolation

Polynomial Interpolation is the process of determining a polynomial that exactly fits a set of data points. It involves finding a polynomial of appropriate degree such that the polynomial's values match the given data at specific points. The uniqueness of the interpolating polynomial, when the nodes are distinct, is a fundamental result, and methods like the Lagrange form are commonly used in approximation theory and numerical analysis.

Lagrange Interpolating Polynomial

The Lagrange Interpolating Polynomial is a method for constructing a polynomial that passes exactly through a given set of points. It uses a weighted sum of specially constructed polynomials such that the resulting polynomial takes on the given values at the specified nodes. This formulation guarantees that the interpolation condition is met for each point and is central to numerical interpolation techniques.

Verify that the polynomial $$ p(x)=\sum_{i=0}^k d_i \prod_{j \neq i} \frac{x-r_j}{r_i-r_j} $$ satisfies $p\left(r_i\right)=d_i$ for $i=0,1, \ldots, k$.

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