Example 8
Completion requirements
| Example 8 |
Simplify the rational expression \(\frac{{2r^2 + 15r + 18}}{{3r^2 - 20r + 12}} \div \frac{{3r^2 + 20r + 12}}{{r^2 - 5r - 6}}\). Identify any non-permissible values.
Step 1: Factor all polynomials.
Step 2: Identify the NPVs.
Step 3: Multiply by the reciprocal.
Step 4: Identify any new NPVs.
Step 5: Simplify.
\[\frac{{2r^2 + 15r + 18}}{{3r^2 - 20r + 12}} \div \frac{{3r^2 + 20r +
12}}{{r^2 - 5r - 6}} = \frac{{\left( {2r + 3} \right)\left( {r + 6}
\right)}}{{\left( {3r - 2} \right)\left( {r - 6} \right)}} \div
\frac{{\left( {3r + 2} \right)\left( {r + 6} \right)}}{{\left( {r - 6}
\right)\left( {r + 1} \right)}}\]
Step 2: Identify the NPVs.
\[\begin{align}
3r - 2 &\ne 0 \\
r &\ne \frac{2}{3} \\
\end{align}\]
3r - 2 &\ne 0 \\
r &\ne \frac{2}{3} \\
\end{align}\]
\[\begin{align}
r - 6 &\ne 0 \\
r &\ne 6 \\
\\
\end{align}\]
r - 6 &\ne 0 \\
r &\ne 6 \\
\\
\end{align}\]
\[\begin{align}
r + 1 &\ne 0 \\
r &\ne -1 \\
\\
\end{align}\]
r + 1 &\ne 0 \\
r &\ne -1 \\
\\
\end{align}\]
Step 3: Multiply by the reciprocal.
\[\frac{{\left( {2r + 3} \right)\left( {r + 6} \right)}}{{\left( {3r -
2} \right)\left( {r - 6} \right)}} \div \frac{{\left( {3r + 2}
\right)\left( {r + 6} \right)}}{{\left( {r - 6} \right)\left( {r + 1}
\right)}} = \frac{{\left( {2r + 3} \right)\left( {r + 6}
\right)}}{{\left( {3r - 2} \right)\left( {r - 6} \right)}} \cdot \frac{{\left( {r - 6} \right)\left( {r + 1} \right)}}{{\left( {3r + 2}
\right)\left( {r + 6} \right)}}\]
Step 4: Identify any new NPVs.
\[\begin{align}
3r + 2 &\ne 0 \\
r &\ne -\frac{2}{3} \\
\end{align}\]
3r + 2 &\ne 0 \\
r &\ne -\frac{2}{3} \\
\end{align}\]
\[\begin{align}
r + 6 &\ne 0 \\
r &\ne -6 \\
\\
\end{align}\]
r + 6 &\ne 0 \\
r &\ne -6 \\
\\
\end{align}\]
Step 5: Simplify.
\[\begin{align}
\frac{{\left( {2r + 3} \right)\left( {r + 6} \right)}}{{\left( {3r - 2} \right)\left( {r - 6} \right)}} \div \frac{{\left( {3r + 2} \right)\left( {r + 6} \right)}}{{\left( {r - 6} \right)\left( {r + 1} \right)}} &= \frac{{\left( {2r + 3} \right) {\color{red} \cancel {\color{#444} \left( {r + 6} \right)}}}}{{\left( {3r - 2} \right) {\color{red} \cancel {\color{#444} \left( {r - 6} \right)}}}} \cdot \frac{{{\color{red} \cancel {\color{#444} \left( {r - 6} \right)}}\left( {r + 1} \right)}}{{\left( {3r + 2} \right) {\color{red} \cancel {\color{#444} \left( {r + 6} \right)}}}} \\
&= \frac{{\left( {2r + 3} \right)\left( {r + 1} \right)}}{{\left( {3r - 2} \right)\left( {3r + 2} \right)}},\thinspace r \ne \pm 6, -1, \pm \frac{2}{3} \\
\end{align}\]
\frac{{\left( {2r + 3} \right)\left( {r + 6} \right)}}{{\left( {3r - 2} \right)\left( {r - 6} \right)}} \div \frac{{\left( {3r + 2} \right)\left( {r + 6} \right)}}{{\left( {r - 6} \right)\left( {r + 1} \right)}} &= \frac{{\left( {2r + 3} \right) {\color{red} \cancel {\color{#444} \left( {r + 6} \right)}}}}{{\left( {3r - 2} \right) {\color{red} \cancel {\color{#444} \left( {r - 6} \right)}}}} \cdot \frac{{{\color{red} \cancel {\color{#444} \left( {r - 6} \right)}}\left( {r + 1} \right)}}{{\left( {3r + 2} \right) {\color{red} \cancel {\color{#444} \left( {r + 6} \right)}}}} \\
&= \frac{{\left( {2r + 3} \right)\left( {r + 1} \right)}}{{\left( {3r - 2} \right)\left( {3r + 2} \right)}},\thinspace r \ne \pm 6, -1, \pm \frac{2}{3} \\
\end{align}\]