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Let $f_{−1},f_{0}$, and $f_{1}$ denote the Lagrange polynomials associated with $−1,0$, and 1 respectively. (2.1) Find $f_{−1},f_{0}$, and $f_{1}$ and express each one in standard polynomial form, that is, $a+bx+cx_{2}$ where $a,b$, and $c$ are real numbers. (2.2) Show that $f_{1}(−x)=f_{−1}(x)$ and $f_{−1}(−x)=f_{1}(x)$. (2.3) Show that $f_{0}$ and $f_{−1}+f_{1}$ are even. (2.4) Show $f∈P_{2}(R)$ is even if and only if $f∈span{f_{0},f_{−1}+f_{1}}$.

We know that ,If the data point are then

Lagrange polynomial is defined as:

(2.1) :

We have to find and and express in the standard form .

Here the Lagrange polynomial associated with are