I fit a 4th-order polynomial curve to data in a chart
and chose the option to display the equation. But when
I put that equation into cells in the spreadsheet, the
evaluated formula does not even come close to the
original Y values, even though I am usin the same X
values. I wonder if it is due to round-off error in
the displayed equation.

How can I get the exact coefficients of a trend line
-- specifically a polynomial of any order -- into a
spreadsheet?

PS: Is it mathematically true that there is always an
N-order polynomial that exactly fits N+1 data points?
That has been my experience so far.