If a polynomial of order n or less passes thru (n+1) points, it is unique!


Given n+1 (x,y) data pairs, with all x values being unique, then a polynomial of order n or less passes thru the (n+1) data points. How can we prove that this polynomial is unique?

I am going to show you the proof for a particular case and you can extend it to polynomials of any order n.

Lets suppose you are given three data points (x1,y1), (x2,y2), (x3,y3) where x1 \neq x2 \neq x3.

Then if a polynomial P(x) of order 2 or less passes thru the three data points, we want to show that P(x) is unique.

We will prove this by contradiction.

Let there be another polynomial Q(x) of order 2 or less that goes thru the three data points. Then R(x)=P(x)-Q(x) is another polynomial of order 2 or less. But the value of P(x) and Q(x) is same at the three x-values of the data points x1, x2, x3. Hence R(x) has three zeros, at x=x1, x2 and x3.

But a second order polynomial only has two zeros; the only case where a second order polynomial can have three zeros is if R(x) is identically equal to zero, and hence have infinite zeros. Since R(x)=P(x)-Q(x), and R(x) \equiv 0, then P(x) \equiv Q(x). End of proof.

But how do you know that a second oder polynomial with three zeros is identically zero.

R(x) is of the form a0+a1*x+a2*x^2 and has three zeros, x1, x2, x3. Then it needs to satisfy the following three equations

a0+a1*x1+a2*x1^2=0

a0+a1*x2+a2*x2^2=0

a0+a1*x3+a2*x3^2=0

The above equations have the trivial solution a0=a1=a2=0 as the only solution if

det(1 x1 x1^2; 1 x2 x2^2; 1 x3 x3^2)\neq0.

That is in fact the case as

det(1 x1 x1^2; 1 x2 x2^2; 1 x3 x3^2) = (x1-x2)*(x2-x3)*(x3-x1),

and since x1 \neq x2 \neq x3, the

det(1 x1 x1^2; 1 x2 x2^2; 1 x3 x3^2) \neq0

So the only solution is a0=a1=a2=0 making R(x) \equiv 0

This post brought to you by Holistic Numerical Methods: Numerical Methods for the STEM undergraduate at http://numericalmethods.eng.usf.edu

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Autar Kaw

Autar Kaw (http://autarkaw.com) is a Professor of Mechanical Engineering at the University of South Florida. He has been at USF since 1987, the same year in which he received his Ph. D. in Engineering Mechanics from Clemson University. He is a recipient of the 2012 U.S. Professor of the Year Award. With major funding from NSF, he is the principal and managing contributor in developing the multiple award-winning online open courseware for an undergraduate course in Numerical Methods. The OpenCourseWare (nm.MathForCollege.com) annually receives 1,000,000+ page views, 1,000,000+ views of the YouTube audiovisual lectures, and 150,000+ page views at the NumericalMethodsGuy blog. His current research interests include engineering education research methods, adaptive learning, open courseware, massive open online courses, flipped classrooms, and learning strategies. He has written four textbooks and 80 refereed technical papers, and his opinion editorials have appeared in the St. Petersburg Times and Tampa Tribune.

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