Sums and Differences of Rational Expressions

$\blacktriangleright$ To combine rational expressions over a common denominator

1.
Enter the expression in mathematics mode.

2.
Leave the insertion point in the expression.

3.
Apply Simplify (or click itbpF0.3035in0.3035in0.0701insimplify.wmf) or Factor.

$\blacktriangleright$ Simplify (or Factor)

${\dfrac{{3x^{2}+3x}}{{8x^{2}+7}}}$ + ${\dfrac{{5x^{2}+3}}{{2x^{2}+x+7}}}$ =  ${\dfrac{{%
46x^{4}+9x^{3}+83x^{2}+21x+21}}{{\left( 8x^{2}+7\right) \left(
2x^{2}+x+7\right) }}}$

${\dfrac{{\left( 3x^{2}+3x\right) ^{3}}}{{2x-5}}}$ - ${\dfrac{{5x^{2}+3}}{{2x^{2}+x+7}}}$

         =  ${\dfrac{{%
54x^{8}+189x^{7}+432x^{6}+702x^{5}+594x^{4}+179x^{3}+25x^{2}-6x+15}}{{\left(
2x-5\right) \left( 2x^{2}+x+7\right) }}}$

${\dfrac{{3x^{2}+4}}{{5x^{3}-3x}}}$ + ${\dfrac{{4x+1}}{{x^{2}-1}}}$ =  ${\dfrac{{%
23x^{4}-11x^{2}-4+5x^{3}-3x}}{{x\left( 5x^{2}-3\right) \left( x^{2}-1\right) }}}$

        or   = ${\dfrac{{23x^{4}-11x^{2}-4+5x^{3}-3x}}{{x\left(
5x^{2}-3\right) \left( x-1\right) \left( x+1\right) }}}$

The feature of computing in placeDM1-3.tex#Computing in place (2) is useful for manipulating polynomials.


\begin{example}
To change a factored denominator to expanded form,
\par
enter t...
...$\frac{x+2}{x+1}+\frac{3x}{x-1}=2\frac{2x^{2}+2x-1}{x^{2}-1}$%
.
\end{example}


\begin{example}
Computing in place, used together with the editing features of ...
...( x-1\right) }=\,2\frac{2x^{2}+2x-1}{x^{2}-1}
\end{displaymath}
\end{example}

You can find the standard form for the equation of a circle by ``completing the square.'' You can take advantage of computing in place for this computation also.


\begin{example}
Find the center and radius of the circle $\,x^{2}-6x+18+y^{2}+1...
...ter of the circle is $(3,-5)$\ and the radius is $\sqrt{16}=\,4$.
\end{example}