A rational expression is a fraction whose numerator and denominator are polynomials.
Simplifying a rational expression means factoring both parts and canceling common factors.
The most important warning is that canceled denominator factors still create domain restrictions.
1. Rational expressions
A rational expression has the form:
\[
\begin{aligned}
\frac{P(x)}{Q(x)}
\end{aligned}
\]
where \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x)\neq 0\).
2. The simplification rule
You may cancel a factor only when it is a factor of the whole numerator and the whole denominator.
\[
\begin{aligned}
\frac{A(x)C(x)}{B(x)C(x)}
&=
\frac{A(x)}{B(x)},
\quad C(x)\neq 0.
\end{aligned}
\]
The condition \(C(x)\neq 0\) is important. It becomes part of the domain restriction.
3. Worked example
Simplify:
\[
\begin{aligned}
\frac{x^2-9}{x^2+6x+9}.
\end{aligned}
\]
Factor the numerator using the difference of squares:
\[
\begin{aligned}
x^2-9
&=
x^2-3^2\\
&=
(x-3)(x+3).
\end{aligned}
\]
Factor the denominator as a perfect-square trinomial:
\[
\begin{aligned}
x^2+6x+9
&=
x^2+2\cdot 3\cdot x+3^2\\
&=
(x+3)^2.
\end{aligned}
\]
Substitute the factored forms:
\[
\begin{aligned}
\frac{x^2-9}{x^2+6x+9}
&=
\frac{(x-3)(x+3)}{(x+3)^2}.
\end{aligned}
\]
Cancel one common factor \(x+3\):
\[
\begin{aligned}
\frac{(x-3)(x+3)}{(x+3)^2}
&=
\frac{x-3}{x+3}.
\end{aligned}
\]
The simplified expression is:
\[
\begin{aligned}
\boxed{
\frac{x-3}{x+3}
}.
\end{aligned}
\]
4. Domain restrictions
The original denominator was:
\[
\begin{aligned}
x^2+6x+9
&=
(x+3)^2.
\end{aligned}
\]
The denominator is zero when:
\[
\begin{aligned}
x+3&=0\\
x&=-3.
\end{aligned}
\]
Therefore, the complete answer is:
\[
\begin{aligned}
\boxed{
\frac{x^2-9}{x^2+6x+9}
=
\frac{x-3}{x+3},
\quad x\neq -3
}.
\end{aligned}
\]
Even though one \(x+3\) factor was canceled, the original expression was still undefined at \(x=-3\).
5. Canceling factors, not terms
This is valid:
\[
\begin{aligned}
\frac{(x-2)(x+5)}{(x-2)(x+1)}
&=
\frac{x+5}{x+1},
\quad x\neq 2,\ -1.
\end{aligned}
\]
But this is not valid:
\[
\begin{aligned}
\frac{x+5}{x+1}
\neq
\frac{5}{1}.
\end{aligned}
\]
The \(x\)-terms are not common factors of the whole numerator and denominator.
6. Holes and vertical asymptotes
A canceled denominator factor usually creates a hole in the graph.
A denominator factor that remains after simplification usually creates a vertical asymptote.
For the example:
\[
\begin{aligned}
\frac{x^2-9}{x^2+6x+9}
&=
\frac{x-3}{x+3},
\quad x\neq -3.
\end{aligned}
\]
The factor \(x+3\) was canceled once, but one \(x+3\) still remains in the denominator.
Therefore \(x=-3\) is still excluded.
7. Formula summary
The table below uses plain text formulas in table cells to avoid raw LaTeX rendering problems.
8. Common mistakes
- Canceling terms instead of factors.
- Canceling before factoring the whole numerator and denominator.
- Forgetting domain restrictions from canceled denominator factors.
- Reporting only the simplified expression without excluded values.
- Changing the value of the expression by canceling part of a sum.
- Forgetting repeated denominator factors such as \((x+3)^2\).
Key idea: factor first, cancel only common factors, and always keep restrictions from the original denominator.
