I suggest you use a tabularx environment instead of the basic tabular environment. Moreover, I think the nine data columns should have equal widths. Use a \settowidth instruction to calculate the required width of the tabularx. You may also want to get rid of all vertical lines, to give the table a more "open" look.

\documentclass{amsart}
\usepackage{bm,booktabs,tabularx}
\newcolumntype{C}{>{\centering\arraybackslash}X}
\newcommand\myheader{\bfseries Force in Newton required to move an object $\bm x$ meters}
\newlength\mylength
\settowidth{\mylength}{\myheader}
\begin{document}
\noindent
\begin{tabularx}{\mylength}{@{} l *{9}{C} @{}}
\toprule
\multicolumn{10}{@{}c@{}}{\myheader} \\ % re-use the \myheader macro
\midrule
$x$ & 4 & 6 & 8 & 10 & 12 & 14 & 16 & 18 & 20 \\
$F(x)$ & 5 & 5.8 & 7 & 8.8 & 9.6 & 8.2 & 6.7 & 5.2 & 4.1 \\
\bottomrule
\end{tabularx}
\end{document}
Addendum to address the OP's follow-up comment: If you do want the vertical bars, you can't use the line-drawing macros of the booktabs package. Instead, I suggest you insert (typographic) struts to provide a bit more vertical whitespace.

\documentclass{amsart}
\usepackage{bm,tabularx}
\newcolumntype{C}{>{\centering\arraybackslash}X}
\newcommand\myheader{\bfseries Force in Newton required to move an object $\bm x$ meters}
\newlength\mylength
\settowidth{\mylength}{\myheader}
\addtolength{\mylength}{2\tabcolsep}
\addtolength{\mylength}{2\arrayrulewidth}
%% define a few struts
%% (from code by Claudio Beccari in TeX and TUG News, Vol. 2, 1993)
\newcommand\Tstrut{\rule{0pt}{2.9ex}} % "top" strut
\newcommand\Bstrut{\rule[-1.2ex]{0pt}{0pt}} % "bottom" strut
\newcommand\TBstrut{\Tstrut\Bstrut} % "top-and-bottom" strut
\begin{document}
\noindent
\begin{tabularx}{\mylength}{ |l| *{9}{C|} }
\hline
\multicolumn{10}{|c|}{\myheader\TBstrut} \\ % re-use the \myheader macro
\hline
$x$\TBstrut & 4 & 6 & 8 & 10 & 12 & 14 & 16 & 18 & 20 \\
\hline
$F(x)$\TBstrut & 5 & 5.8 & 7 & 8.8 & 9.6 & 8.2 & 6.7 & 5.2 & 4.1 \\
\hline
\end{tabularx}
\end{document}
\myheaderwith\newcommand\myheader{\bfseries Force in Newton... $\bm x$ meters}? Could you have avoided this and just had the commands\newlength\mylengthand\settowidth{\mylength}{\bfseries Force in Newton... $\bm x$ meters}? – Adelyn Feb 03 '16 at 22:38$\bm x$just a more efficient code for\boldmath$x$\unboldmath? – Adelyn Feb 03 '16 at 22:39\myheaderis so that the string\bfseries Force in Newton required to move an object $\bm x$ metersneedn't be typed twice (first when the width of the string is measured, and later when it's typset). And, should you choose later on to use a different header string, you'll only need to modify it once. The macro\bmnot only is more efficient, it also avoids the bad spacing that arises from...\boldmath$x$\unboldmath.... – Mico Feb 03 '16 at 22:48f(x)andxin the header, would you code something like\bfseries Force $bm f(x)$ in Newton required to move an object $\bm x$ meters? Is\newcolumntype{C}{>{\centering\arraybackslash}X}in the preamble used? – Adelyn Feb 03 '16 at 22:58\bm, notbm(note the backslash character that precedesbm). I believe that in physics, the universally used symbol for "force" isF, so I'd writeF(x), notf(x). Finally, the instruction\newcolumntype{C}{>{\centering\arraybackslash}X}gets used in the final argument of thetabularxenvironment setup, viz., in the string|l| *{9}{C|}. – Mico Feb 03 '16 at 23:02tabularxpackage in case I want to include other tables in a file. May you suggest a web site containing its manual? If it is not too inconvenient, please explain the syntax of\newcolumntype{C}{>{\centering\arraybackslash}X}and what it does to{ |l| *{9}{C| }. – Adelyn Feb 04 '16 at 15:21\newcolumntypeinstruction creates a column type namedC, which centers its contents (because of\centering) but otherwise inherits the properties of theXcolumn type, which is defined by thetabularxpackage. (See the manual of thearraypackage for an explanation of what\arraybackslashdoes.) And, please read the manual of the tabularx package. The string*{9}{C|}is shorthand for "9columns of typeC, each one followed by|(a vertical bar)". – Mico Feb 04 '16 at 15:32