[Math] Prove there exists a unique $n$-th degree polynomial that passes through $n+1$ points in the plane

Question

I know given two points in the plane $(x_1,y_1)$ and $(x_2,y_2)$ there exists a unique 1^st degree (linear) polynomial that passes through those points. We all learned in Algebra how to find the slope between those points and then calculate the y-intercept.

To take it down a notch, given the point $(a,b)$, the unique 0^th degree polynomial that passes through it is $y=b$.

My conjecture is that given three points $(x_1,y_1)$, $(x_2,y_2)$, and $(x_3,y_3)$, there exists a 2^nd degree (quadratic) polynomial that passes through these points, and furthermore, that polynomial is unique. I wonder, how would one determine the equation of this quadratic?

If my conjecture is correct, a corollary would be the generalization that given any $\left(n+1\right)$ points in the plane, there exists one unique $n$^th degree polynomial that passes through those points.

Please prove, or disprove with a counter-example.

Further Readings:

• 658789 is a related question but I'm not sure if it's exactly what I'm looking for.
• This very cool interactive web app lets you drag points around and shows the polynomial that goes through them

Answer 1

Assume you have given $n+1$ points $(x_1,y_1),\cdots, (x_{n+1},y_{n+1}).$ (Of course, $x_i\ne x_j$ if $i\ne j.$) A polynomial of degree $n$ is of the form $p_n(x)=a_nx^n+\cdots+a_1x+a_0.$ To study the existence and uniqueness of such a polynomial consider the system of linear equations:

$$ \left\{\begin{array}{ccc} a_nx_1^n+a_{n-1}x_1^{n-1}\cdots+a_1x_1+a_0 & =& y_1\\ \vdots & &\\ a_nx_{n+1}^n+a_{n-1}x_{n+1}^{n-1}\cdots+a_1x_{n+1}+a_0 & =& y_{n+1} \end{array}\right. $$

We write the system as

$$ \begin{pmatrix}x_1^n & x_1^{n-1} &\cdots & x_1 & 1 \\ \vdots & \vdots & \ddots & \vdots \\ x_{n+1}^n & x_{n+1}^{n-1}& \cdots & x_{n+1} & 1\end{pmatrix} \begin{pmatrix} a_n \\ \vdots \\ a_0 \end{pmatrix}=\begin{pmatrix} y_1 \\ \vdots \\ y_{n+1} \end{pmatrix} $$

Since the matrix of coefficients of the system is non singular (it is a Vandermonde matrix (see Vandermonde)) the system has a unique solution, that is, there exists one polynomial of degree $n$ through the $n+1$ given points, and it is unique.