Chapter 3.3, Problem 54E

For Exercises 44 through 61, consider the problem of fitting a cubic through m given points $P_1\left(x_1,y_1\right),\mathrm{...},P_m\left(x_m,y_m\right)$ in the plane. A cubic is a curve in ${ℝ}^2$ that can be described by an equation of the form $f\left(x,y\right)=c_1+c_{2}x+c_{3}y+c_4{x}^2+c_{5}xy+c_6{y}^2+c_7{x}^3+c_8{x}^{2}y+c_{9}x{y}^2+c_{10}{y}^3=0$ , where at least one of the coefficients $c_i$ is nonzero. If k is any nonzero constant, then the equations $f\left(x,y\right)=0$ and $kf\left(x,y\right)=0$ definethe same cubic.

54. $\left(0,0\right),\left(1,0\right),\left(2,0\right),\left(0,1\right),\left(1,1\right),\left(2,1\right),\left(0,2\right),\left(1,2\right),\left(2,2\right)$