Just as a proof of concept, I set up a Numbers spreadsheet to find zeroes of a 5th-degree polynomial. One table, named "Guesses", contained in its first row a current guess, and computed in its second row a next guess. Column 1 of the "Guesses" table contains labels, so the current and next guess are actually in column 2.

How the next guess is computed isn't particularly germane here, but since the derivative of a polynomial is easy to evaluate I used Newton-Raphson. Since this was just proof of concept, I didn't bother defending against degenerate cases, like when the slope is zero.) For problems where the derivative is intractable, the secant method could be used instead, or even bisection.

Another table contained the coefficients, plus some workspace to evaluate the the polynomial and its derivative at the current guess.

Then I wrote the following AppleScript:
Code:
tell application "Numbers"
	tell document 1
		tell sheet 1
			tell table "Guesses"
				set value of cell 2 of row 1 to value of cell 2 of row 2
			end tell
		end tell
	end tell
end tell

Put in an initial guess, and keep clicking on the "Run" button in AppleScript Editor" to watch the spreadsheet converge on the solution. Works pretty well in the test case.