Simplification of Rational Expressions
Given the rational expression:
[tex]\frac{n^4-10n^2+24}{n^4-9n^2+18}[/tex]
Simplify and state the restriction for the variable n.
Let's work on the numerator and denominator independently. Factoring the numerator:
[tex]\begin{gathered} n^4-10n^2+24=n^4-4n^2-6n^2+24 \\ n^4-10n^2+24=n^2(n^2-4)-6(n^2-4) \\ n^4-10n^2+24=(n^2-6)\mleft(n^2-4\mright) \end{gathered}[/tex]
The denominator can be factored in a similar way:
[tex]\begin{gathered} n^4-9n^2+18=n^4-3n^2-6n^2+18 \\ n^4-9n^2+18=n^2(n^2-3)-6(n^2-3) \\ n^4-9n^2+18=\mleft(n^2-3\mright)(n^2-6) \end{gathered}[/tex]
Thus, rewriting the expression:
[tex]\frac{n^4-10n^2+24}{n^4-9n^2+18}=\frac{(n^2-4)(n^2-6)}{(n^2-3)(n^2-6)}[/tex]
Before simplifying, we must state the restrictions for the variable. The denominator cannot be 0, thus:
[tex]\begin{gathered} n^2-3\ne0\Rightarrow n\ne\pm\sqrt[]{3} \\ n^2-6\ne0\Rightarrow n\ne\pm\sqrt[]{6} \end{gathered}[/tex]
Now simplify:
[tex]\frac{n^4-10n^2+24}{n^4-9n^2+18}=\frac{(n^2-4)}{(n^2-3)}[/tex]
Combining the final expression with the restrictions, we stick with choice a.
