SOLUTION: How do you find the polynomial whose graph goes through these points? (2,10), (-1,-4), (0,2), and (1,-2). Thanks

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Question 1102896: How do you find the polynomial whose graph goes through these points?
(2,10), (-1,-4), (0,2), and (1,-2).
Thanks
Found 2 solutions by greenestamps, Fombitz:
Answer by greenestamps(13215) (Show Source):
You can put this solution on YOUR website!

n points can be fitted with a polynomial of degree (n-1). With 4 given points, you need a polynomial of degree 3:

Simply make 4 equations in the unknown coefficients a, b, c, and d using the coordinates of the given points.
(2,10) --> [1]
(-1,-4) --> [2]
(0,2) --> [3]
(1,-2) --> [4]

Use [3] to simplify the others:
[5]
[6]
[7]

Add [6] and [7] to solve for b:

[8]

Use [8] to simplify [5] and [7]:

[9]

[10]

Solve [9] and [10] by elimination to solve for a and c:

The polynomial is

Here is the graph...

Answer by Fombitz(32388) (Show Source):
You can put this solution on YOUR website!
You can use EXCEL to do a curve fit.
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You can use linear algebra to solve a system of equations using a general form of the polynomial.

So then,
(2,10):

1.
(-1,-4):

2.
(0,2):

3.
(1,-2):

4.
Then substitute eq. 3 into the others,

5.
and

6.
and

7.
Continuing, add eq. 5 and eq. 7 to eq. 6 to eliminate C.

8.
and

9.
So then working backwards,
from eq. 8,

and from eq. 7,

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Of course, it matches the EXCEL curve fit.