Exercise 2.2.21

For the closest cubic $b=C+\mathit{Dt}+E{t}^2+F{t}^3$, we have $\mathit{Ax}=b$, i.e. $\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 3\hfill & \hfill 9\hfill & \hfill 27\hfill \\ \hfill 1\hfill & \hfill 4\hfill & \hfill 16\hfill & \hfill 64\hfill \end{array}\right]\left[\begin{array}{c}\hfill C\hfill \\ \hfill D\hfill \\ \hfill E\hfill \\ \hfill F\hfill \end{array}\right]=\left[\begin{array}{c}\hfill 0\hfill \\ \hfill 8\hfill \\ \hfill 8\hfill \\ \hfill 20\hfill \end{array}\right]$,

Solve by elimination we have $C=0,D=\frac{47}{3},E=-\frac{28}{3},F=\frac{5}{3}$.