en forma matricial $$ \begin{equation*} u = \alpha_{0} + \alpha_{1} x + \alpha_{2} x^{2} = \begin{bmatrix} 1 & x & x^{2} \end{bmatrix} \begin{bmatrix} \alpha_{0} \\ \alpha_{1} \\ \alpha_{2} \end{bmatrix} \end{equation*} $$
reemplazando los puntos \( x_{1} \), \( x_{2} \) y \( x_{3} \) $$ \begin{align*} \alpha_{0} + \alpha_{1} x_{1} + \alpha_{2} x_{1}^{2} &= u_{1} \\ \alpha_{0} + \alpha_{1} x_{2} + \alpha_{2} x_{2}^{2} &= u_{2} \\ \alpha_{0} + \alpha_{1} x_{3} + \alpha_{2} x_{3}^{2} &= u_{3} \end{align*} $$
en forma matricial $$ \begin{align*} \begin{bmatrix} 1 & x_{1} & x_{1}^{2} \\ 1 & x_{2} & x_{2}^{2} \\ 1 & x_{3} & x_{3}^{2} \end{bmatrix} \begin{bmatrix} \alpha_{0} \\ \alpha_{1} \\ \alpha_{2} \end{bmatrix} = \begin{bmatrix} u_{1} \\ u_{2} \\ u_{3} \end{bmatrix} \end{align*} $$
resolviendo $$ \begin{equation*} \begin{bmatrix} \alpha_{0} \\ \alpha_{1} \\ \alpha_{2} \end{bmatrix} = \begin{bmatrix} \frac{x_{2} x_{3}}{x_{1}^{2} - x_{1} x_{2} - x_{1} x_{3} + x_{2} x_{3}} & -\frac{x_{1}x_{3}}{x_{1} x_{2} - x_{1} x_{3} - x_{2}^{2} + x_{2} x_{3}} & \frac{x_{1} x_{2}}{x_{1} x_{2} - x_{1} x_{3} - x_{2} x_{3} + x_{3}^{2}} \\ -\frac{x_{2} + x_{3}}{x_{1}^{2} - x_{1} x_{2} - x_{1} x_{3} + x_{2} x_{3}} & \frac{x_{1} + x_{3}}{x_{1} x_{2} - x_{1} x_{3} - x_{2}^{2} + x_{2} x_{3}} & -\frac{x_{1} + x_{2}}{x_{1} x_{2} - x_{1} x_{3} - x_{2} x_{3} + x_{3}^{2}} \\ \frac{1}{x_{1}^{2} - x_{1} x_{2} - x_{1} x_{3} + x_{2} x_{3}} & -\frac{1}{x_{1} x_{2} - x_{1} x_{3} - x_{2}^{2} + x_{2} x_{3}} & \frac{1}{x_{1} x_{2} - x_{1} x_{3} - x_{2} x_{3} + x_{3}^{2}} \end{bmatrix} \begin{bmatrix} u_{1} \\ u_{2} \\ u_{3} \end{bmatrix} \end{equation*} $$
reemplazando las incógnitas $$ \begin{align*} u &= \begin{bmatrix} 1 & x & x^{2} \end{bmatrix} \begin{bmatrix} \alpha_{0} \\ \alpha_{1} \\ \alpha_{2} \end{bmatrix} \\ &= \begin{bmatrix} 1 & x & x^{2} \end{bmatrix} \begin{bmatrix} \frac{x_{2} x_{3}}{x_{1}^{2} - x_{1} x_{2} - x_{1} x_{3} + x_{2} x_{3}} & -\frac{x_{1}x_{3}}{x_{1} x_{2} - x_{1} x_{3} - x_{2}^{2} + x_{2} x_{3}} & \frac{x_{1} x_{2}}{x_{1} x_{2} - x_{1} x_{3} - x_{2} x_{3} + x_{3}^{2}} \\ -\frac{x_{2} + x_{3}}{x_{1}^{2} - x_{1} x_{2} - x_{1} x_{3} + x_{2} x_{3}} & \frac{x_{1} + x_{3}}{x_{1} x_{2} - x_{1} x_{3} - x_{2}^{2} + x_{2} x_{3}} & -\frac{x_{1} + x_{2}}{x_{1} x_{2} - x_{1} x_{3} - x_{2} x_{3} + x_{3}^{2}} \\ \frac{1}{x_{1}^{2} - x_{1} x_{2} - x_{1} x_{3} + x_{2} x_{3}} & -\frac{1}{x_{1} x_{2} - x_{1} x_{3} - x_{2}^{2} + x_{2} x_{3}} & \frac{1}{x_{1} x_{2} - x_{1} x_{3} - x_{2} x_{3} + x_{3}^{2}} \end{bmatrix} \begin{bmatrix} u_{1} \\ u_{2} \\ u_{3} \end{bmatrix} \\ &= \begin{bmatrix} \frac{x_{2} x_{3}}{x_{1}^{2} - x_{1} x_{2} -x_{1} x_{3} + x_{2} x_{3}} - \frac{x_{2} + x_{3}}{x_{1}^{2} - x_{1} x_{2} - x_{1} x_{3} + x_{2} x_{3}} x + \frac{1}{x_{1}^{2} - x_{1} x_{2} - x_{1} x_{3} + x_{2} x_{3}} x^{2} \\ - \frac{x_{1} x_{3}}{x_{1} x_{2} - x_{1} x_{3} - x_{2}^{2} + x_{2} x_{3}} + \frac{x_{1} + x_{3}}{x_{1} x_{2} - x_{1} x_{3} - x_{2}^{2} + x_{2} x_{3}} x - \frac{1}{x_{1} x_{2} - x_{1} x_{3} - x_{2}^{2} + x_{2} x_{3}} x^{2} \\ \frac{x_{1} x_{2}}{x_{1} x_{2} - x_{1} x_{3} - x_{2} x_{3} + x_{3}^{2}} - \frac{x_{1} + x_{2}}{x_{1} x_{2} - x_{1} x_{3} - x_{2} x_{3} + x_{3}^{2}} x + \frac{1}{x_{1} x_{2} - x_{1} x_{3} - x_{2} x_{3} + x_{3}^{2}} x^{2} \end{bmatrix}^{\mathrm{T}} \begin{bmatrix} u_{1} \\ u_{2} \\ u_{3} \end{bmatrix} \\ &= \begin{bmatrix} N_{1} \\ N_{2} \\ N_{3} \end{bmatrix}^{\mathrm{T}} \begin{bmatrix} u_{1} \\ u_{2} \\ u_{3} \end{bmatrix} \end{align*} $$