Example 2
Completion requirements
| Example 2 |
Simplify the expression \(\frac{{6x^2y - 2xy^2 }}{{4x^2y + 6xy^2 }}\). Identify the non-permissible values.
Step 1: Factor each polynomial.
Only factors common to the numerator and denominator can be eliminated. Single terms connected to other terms by addition and subtraction cannot be simplified or eliminated.
Step 2: Determine the NPVs.
Step 3: Simplify by removing factors found in both the numerator and denominator.
The simplified expression is \(\frac{3x - y}{2x + 3y}, \thinspace x \ne 0,\thinspace y \ne 0, \thinspace x \ne -\frac{3}{2}y, \thinspace y \ne -\frac{2}{3}x\).
\[\frac{{6x^2y - 2xy^2}}{{4x^2y + 6xy^2}} = \frac{{2xy\left( {3x - y} \right)}}{{2xy\left( {2x + 3y} \right)}}\]
Only factors common to the numerator and denominator can be eliminated. Single terms connected to other terms by addition and subtraction cannot be simplified or eliminated.
Step 2: Determine the NPVs.
\(\begin{align}
x &\ne 0 \\
y &\ne 0 \\
y &\ne -\frac{2}{3}x \\
x &\ne -\frac{3}{2}y \\
\end{align}\)
x &\ne 0 \\
y &\ne 0 \\
y &\ne -\frac{2}{3}x \\
x &\ne -\frac{3}{2}y \\
\end{align}\)
Step 3: Simplify by removing factors found in both the numerator and denominator.
\[\frac{{\color{red} \cancel {\color{#444}{2xy}}} \left( {3x - y}
\right)}{{\color{red}\cancel {\color{#444}{2xy}}}\left( {2x + 3y} \right)} = \frac{{3x
- y}}{{2x + 3y}}\]
The simplified expression is \(\frac{3x - y}{2x + 3y}, \thinspace x \ne 0,\thinspace y \ne 0, \thinspace x \ne -\frac{3}{2}y, \thinspace y \ne -\frac{2}{3}x\).