Finding Polynomials

Date: 04/29/97 at 01:16:33
From: Elizabeth
Subject: Algebra I - Find the polynomial

I need to know what algebric equation fits the following data:

     x   y
     --  --
     1    6
     2   13
     3   21
     4   30
     5   40
     6   51

Thank you!

Date: 04/29/97 at 15:03:47
From: Doctor Rob
Subject: Re: Algebra I - Find the polynomial

The usual way to work this kind of problem is to look at the 
difference of successive values of y:

   x:  1   2   3   4   5   6
   y:  6  13  21  30  40  51
         7   8   9  10  11

Next, find the differences of the differences:

   x:  1   2   3   4   5   6
   y:  6  13  21  30  40  51
         7   8   9  10  11
           1   1   1   1

These are a constant (1), and so the equation is going to look like:

   y = a*x^2 + b*x + c (second degree)

To find a, b, and c, use the first three (x,y) pairs:

    6 = a*1^2 + b*1 + c =   a +   b + c
   13 = a*2^2 + b*2 + c = 4*a + 2*b + c
   21 = a*3^2 + b*3 + c = 9*a + 3*b + c

Here you have three simultaneous equations in three unknowns a, b, and 
c. Subtract the second from the third, and the first from the second:

   6 =   a + b + c
   7 = 3*a + b
   8 = 5*a + b

Now subtract the new second from the new third:

   6 =   a + b + c
   7 = 3*a + b
   1 = 2*a

Now it is easy to figure out a from the last equation.  Using that,
figure out b from the previous one, and using both, figure out c from
the one before that.  Once you know a, b, and c, you know what y is
as a polynomial in x.

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/