Fractions

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Fractions

Now consider the following polynomial fraction:

$\begin{displaymath}\frac{x^3-y^3}{x^2+x-y-y^2} = \frac{y^2+xy+x^2}{y+1+x} \end{displaymath}$

The expression on the right is obtained by extracting the greatest common divider, which in this case is y - x, from both the numerator and the denominator. In Maxima this is done by calling ezgcd:

$\begin{figure} \begin{tex2html_preform}\begin{verbatim}(C81) ezgcd(x^3 - y^3, x^... ...Y - X Y - X , - Y - X - 1] (C82)\end{verbatim}\end{tex2html_preform}\end{figure}$

Function ezgcd returns a list. Members of this list can be accessed as follows:

$\begin{figure} \begin{tex2html_preform}\begin{verbatim}(C93) a : ezgcd(x^3 - y^3... ... X, - Y - X Y - X , - Y - X - 1]\end{verbatim}\end{tex2html_preform}\end{figure}$

There are three terms in the list returned by Maxima. The first term is the greatest common divider, and the second and the third term are the result of dividing the arguments of ezgcd by the greatest common divider. We can verify this as follows:

$\begin{figure} \begin{tex2html_preform}\begin{verbatim}(C94) a[1] * a[2]; 2 2 (... ...d(%); 2 2 (D97) - Y - Y + X + X\end{verbatim}\end{tex2html_preform}\end{figure}$

We can use the elements of the list returned by ezgcd in order to construct a simplified (or normalised) fraction:

$\begin{figure} \begin{tex2html_preform}\begin{verbatim}(C98) b : -a[2] / -a[3]; ... ...) ------------- Y + X + 1 (C99)\end{verbatim}\end{tex2html_preform}\end{figure}$

If you are in a hurry you can accomplish the same simply by saying:

$\begin{figure} \begin{tex2html_preform}\begin{verbatim}(C21) factor((x^3 - y^3) ... ...) ------------- Y + X + 1 (C22)\end{verbatim}\end{tex2html_preform}\end{figure}$

Maple has a function, called normal, which normalises fractions:

$\begin{figure} \begin{tex2html_preform}\begin{verbatim}> normal((x^3 - y^3)/(x^2... ...+ y ------------- x + 1 + y>\end{verbatim}\end{tex2html_preform}\end{figure}$

But factor will work too:

$\begin{figure} \begin{tex2html_preform}\begin{verbatim}> factor((x^3 - y^3) / (x... ...+ y ------------- x + 1 + y>\end{verbatim}\end{tex2html_preform}\end{figure}$

And there is also a similar function in Mathematica, called Factor:

$\begin{figure} \begin{tex2html_preform}\begin{verbatim}In[2]:= Factor [ (x^3 - y... ...------------ 1 + x + yIn[3]:=\end{verbatim}\end{tex2html_preform}\end{figure}$

Next: Substitutions Up: Algebra Previous: Trigonometry

Zdzislaw Meglicki
2001-02-26