(Solved): Suppose experimental data are represented by a set of points in the plane. An interpolating polyno ...

Suppose experimental data are represented by a set of points in the plane. An interpolating polynomial for the data is a polynomial whose graph passes through every point. In scientific work, such a polynomial can be used, for example, to estimate values between the known data points. Another use is to create curves for graphical images on a computer screen. One method for finding an interpolating polynomial is to solve a system of linear equations. Find the interpolating polynomial \( p(t)=a_{0}+a_{1} t+a_{2} t^{2} \) for the data \( (1,15),(2,19) \), \( (3,21) \). That is, find \( a_{0}, a_{1} \), and \( a_{2} \) such that the following is true. \[ \begin{array}{l} a_{0}+a_{1}(1)+a_{2}(1)^{2}=15 \\ a_{0}+a_{1}(2)+a_{2}(2)^{2}=19 \\ a_{0}+a_{1}(3)+a_{2}(3)^{2}=21 \end{array} \] Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The interpolating polynomial is \( p(t)= \) B. There are infinitely many possible interpolating polynomials. C. There does not exist an interpolating polynomial for the given data.