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Given n Points, find a Polynomial Function...


Date: 6/27/96 at 22:36:47
From: Anonymous
Subject: Given n Points, find a Polynomial Function...

Dr. Math,

    I am having a problem with this question.  Can you please help?

Suppose (x1,y1),(x2,y2),...,(x(n),y(n))  are n points in the plane 
with the x(i)'s all different.  

Find a polynomial f1(x) of degree less than n so that 
	f1(x1) = 1 and  f1(x(j)) = 0    for all j not equal to 1

Find a polynomial F(x) of degree less than n so that the graph of F 
passes through all of the points (x1,y1),(x2,y2),..., (x(n),y(n)).

Thank you.

Richard


Date: 6/28/96 at 13:36:23
From: Doctor Ceeks
Subject: Re: Given n Points, find a Polynomial Function...

Hi,

Let pj(x) = (x-x1)(x-x2)...(x-x(n))/(x-xj).
Let fj(x) = pj(x)/pj(xj).

Then fj(xk) = 1 if j = k and 0 otherwise.

Now let F = y1f1+y2f2+...+y(n)fn.

These are standard constructions.

-Doctor Ceeks,  The Math Forum
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High School Basic Algebra
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