(Solved): In the following exercise we consider using a different form of a polynomial, called Lagrange inter ...

In the following exercise we consider using a different form of a polynomial, called Lagrange interpolating polynomial. Exercise 4. Assuming \( x_{0}, x_{1} \), and \( x_{2} \) are distinct, consider the polynomial \[ P(x)=y_{0} \frac{\left(x-x_{1}\right)\left(x-x_{2}\right)}{\left(x_{0}-x_{1}\right)\left(x_{0}-x_{2}\right)}+y_{1} \frac{\left(x-x_{0}\right)\left(x-x_{2}\right)}{\left(x_{1}-x_{0}\right)\left(x_{1}-x_{2}\right)}+y_{2} \frac{\left(x-x_{0}\right)\left(x-x_{1}\right)}{\left(x_{2}-x_{0}\right)\left(x_{2}-x_{1}\right)} . \] Verify that \( P\left(x_{0}\right)=y_{0}, P\left(x_{1}\right)=y_{1} \), and \( P\left(x_{2}\right)=y_{2} \).