Rational expressions are fractions made from polynomials. Operations on rational expressions follow the same
basic rules as operations on ordinary fractions, but factoring and domain restrictions are especially important.
1. Rational expressions
A rational expression has the form:
\[
\begin{aligned}
\frac{P(x)}{Q(x)},
\quad Q(x)\neq 0.
\end{aligned}
\]
The denominator cannot be zero. These forbidden values are called domain restrictions or excluded values.
2. Addition and subtraction
To add or subtract rational expressions, first build a common denominator:
\[
\begin{aligned}
\frac{A}{B}+\frac{C}{D}
&=
\frac{AD+CB}{BD},
\end{aligned}
\]
\[
\begin{aligned}
\frac{A}{B}-\frac{C}{D}
&=
\frac{AD-CB}{BD}.
\end{aligned}
\]
After combining into one fraction, factor and simplify if possible.
3. Worked addition example
Add:
\[
\begin{aligned}
\frac{x+2}{x-3}
+
\frac{x-1}{x+4}.
\end{aligned}
\]
The common denominator is:
\[
\begin{aligned}
(x-3)(x+4).
\end{aligned}
\]
Rewrite both fractions with that denominator:
\[
\begin{aligned}
\frac{x+2}{x-3}
+
\frac{x-1}{x+4}
&=
\frac{(x+2)(x+4)+(x-1)(x-3)}{(x-3)(x+4)}.
\end{aligned}
\]
Expand the numerator:
\[
\begin{aligned}
(x+2)(x+4)
&=
x^2+6x+8,\\
(x-1)(x-3)
&=
x^2-4x+3.
\end{aligned}
\]
Add:
\[
\begin{aligned}
x^2+6x+8+x^2-4x+3
&=
2x^2+2x+11.
\end{aligned}
\]
Therefore:
\[
\begin{aligned}
\boxed{
\frac{x+2}{x-3}
+
\frac{x-1}{x+4}
=
\frac{2x^2+2x+11}{(x-3)(x+4)}
}.
\end{aligned}
\]
The excluded values are:
\[
\begin{aligned}
x&\neq 3,\\
x&\neq -4.
\end{aligned}
\]
4. Multiplication
To multiply rational expressions, multiply numerator by numerator and denominator by denominator:
\[
\begin{aligned}
\frac{A}{B}\cdot\frac{C}{D}
&=
\frac{AC}{BD}.
\end{aligned}
\]
It is often helpful to factor before multiplying, because common factors may cancel.
Example:
\[
\begin{aligned}
\frac{x^2-9}{x+4}\cdot\frac{x+4}{x-3}.
\end{aligned}
\]
Factor \(x^2-9\):
\[
\begin{aligned}
x^2-9
&=
(x-3)(x+3).
\end{aligned}
\]
Then:
\[
\begin{aligned}
\frac{(x-3)(x+3)}{x+4}
\cdot
\frac{x+4}{x-3}
&=
x+3,
\end{aligned}
\]
but the original restrictions still include \(x\neq -4\) and \(x\neq 3\).
5. Division
Division by a rational expression means multiplying by its reciprocal:
\[
\begin{aligned}
\frac{A}{B}\div\frac{C}{D}
&=
\frac{A}{B}\cdot\frac{D}{C}.
\end{aligned}
\]
In division, there is one extra domain rule: the expression you divide by cannot equal zero.
Therefore \(C\neq 0\) as well as \(B\neq 0\) and \(D\neq 0\).
6. Full simplification
After performing the operation, write the answer as one rational expression:
\[
\begin{aligned}
\frac{\text{new numerator}}{\text{new denominator}}.
\end{aligned}
\]
Then factor the numerator and denominator and cancel only common factors:
\[
\begin{aligned}
\frac{A(x)C(x)}{B(x)C(x)}
&=
\frac{A(x)}{B(x)},
\quad C(x)\neq 0.
\end{aligned}
\]
Canceled factors do not remove original restrictions. They only simplify the expression on the allowed domain.
7. Domain restrictions
For addition, subtraction, and multiplication, excluded values come from the original denominators.
For division, excluded values also come from the numerator of the second rational expression, because the divisor
cannot equal zero.
For example:
\[
\begin{aligned}
\frac{x+1}{x-2}
\div
\frac{x-5}{x+3}
\end{aligned}
\]
has restrictions:
\[
\begin{aligned}
x&\neq 2,\\
x&\neq -3,\\
x&\neq 5.
\end{aligned}
\]
The value \(x=5\) is excluded because the second fraction would equal zero, and division by zero is undefined.
8. Why common denominators work
A common denominator lets two fractions describe pieces of the same whole.
For rational expressions:
\[
\begin{aligned}
\frac{A}{B}
&=
\frac{AD}{BD},
\quad
\frac{C}{D}
=
\frac{CB}{DB}.
\end{aligned}
\]
Now both fractions have denominator \(BD\), so their numerators can be added or subtracted.
9. Formula summary
The table below uses plain text formulas in table cells to avoid raw LaTeX rendering problems.
10. Common mistakes
- Adding numerators and denominators directly.
- Forgetting to build a common denominator for addition or subtraction.
- Dropping parentheses around a numerator during subtraction.
- Dividing without flipping the second fraction.
- Canceling terms instead of full factors.
- Forgetting excluded values from original denominators.
- Forgetting that, in division, the second rational expression cannot equal zero.
Key idea: perform the fraction operation first, simplify by factoring, and keep all original domain restrictions.
