Wrong coefficients in a polynomial fit I am trying to fit data to a fourth-degree polynomial. I tried this in multiple programs (R, Origin Pro, SigmaPlot), all of which give me a polynomial of the form
$ 40000 -2000x + 40x^2 -0.3x^3 + 0.001x^4 $. This doesn't fit the data at all (the y-intercept should be close to zero). All data points are relatively near each other. However, when the programs graphically show the fitted polynomial, it looks like this:

The shown polynomial clearly fits the data quite well and is (!) distinct from the one given above. For example, the y-intercepts don't match at all. Is there something about regression I don't understand or why do all of the programs plot a different polynomial than they return? Could this be an overflow problem?
 A: 
The shown polynomial clearly fits the data quite well and is (!) distinct from the one given above. For example, the y-intercepts don't match at all. Is there something about regression I don't understand or why do all of the programs plot a different polynomial than they return?

Round off errors
The polynomial that they return has coefficients that are rounded off (for display). It is not the polynomial that they use in processing (which uses the full precision).
When I look at the fitted values I do not see why the curve and presented polynomial would not have a y-intercept close to zero.
The y-intercept value of the returned polynomial, taken at x=63 (at least that's where the y-axis crosses in this image), is -36:
$$40992.96617 - 2125.34077 \cdot 63 + 40.79366 \cdot 63^2 - 0.34353 \cdot 63^3  + 0.00107 \cdot 63^4 \approx -36$$
But, that's a reasonably close y-intercept. The roundoff errors in the returned polynomial are gonna give huge effect because these powers of X are very large.
For example, when the last coefficient is 0.001074 instead of 0.00107 then this adds $0.000004* 63^4 \approx 63$ to the value of the intercept with the y-axis at x=63.
Solution
To solve your problem: When you rescale $X$ before the fitting then the roundoff errors in the coefficents will have smller effects. Also you could look into the software how to obtain the returned polynomial with coefficients that have a higher precision.
A: First, the coefficients in your graph are very close to the ones in your text. So it's not clear why you say 

and is (!) distinct from the one given above

It isn't.
In particular, you write that 

the y-intercept should be close to zero

No, it shouldn't. The y-intercept in your figure is 40993 and in your formula it is 40,000. The problem is that your graph does not show the y intercept at all. It shows that y will be close to 0 when X is about 65. 
Then you ask

Is there something about regression I don't understand or why do all
  of the programs plot a different polynomial than they return?

There may be something about regression that you don't understand - it's a huge topic - but this question is not evidence of such misunderstanding (but see below). You have simply misinterpreted the output.  And ... the programs do plot the polynomial they return.
Finally, you ask if this is an overflow problem. No. It isn't.
You could try graphing this with a lower limit of 0 for X. That will show one thing you may not have considered which is the danger of extrapolation (I don't know if you intend to extrapolate). 
Another potential problem is that your x values are clumped. If you tell us what X and Y are, then we might be able to help more with this. 
A: The problem was that the coefficients given by the program were not displayed in full precision. This caused a round-off error which caused the polynomial given by the equation and the polynomial given in the graphic to deviate.
