Subtract rational expressions

Subtracting rational expressions involves interpreting each expression as a quotient defined only where its denominator is nonzero, finding a common denominator—often by factoring and using the least common denominator—and combining equivalent fractions by subtracting their numerators while preserving domain restrictions. The reasoning emphasizes distributing the subtraction across the numerator, avoiding subtraction of denominators, and simplifying the resulting expression; it supports solving rational equations, where excluded values must remain excluded. Partial-fraction decomposition and more abstract rational-function operations are beyond this scope.

Detailed Explanation: Subtract rational expressions

To subtract rational expressions:

State the values that are not allowed by the original denominators.
Factor the denominators.
Find the least common denominator (LCD).
Rewrite each fraction using the LCD.
Subtract the numerators, distributing the minus sign.
Simplify, while keeping the original restrictions.

Example

Simplify

\frac{2x}{x^2-9}-\frac{1}{x-3}.

Step 1: Find the restrictions.

The denominators cannot equal zero:

x^2-9\ne 0 \quad\text{and}\quad x-3\ne 0.

Factor the first denominator:

x^2-9=(x-3)(x+3).

Therefore,

x\ne 3 \quad\text{and}\quad x\ne -3.

These restrictions must remain in the final answer.

Step 2: Find the LCD.

The denominators are $(x-3)(x+3)$ and $x-3$ , so the LCD is

(x-3)(x+3).

Step 3: Rewrite both fractions using the LCD.

The first fraction already has the LCD:

\frac{2x}{(x-3)(x+3)}.

For the second fraction, multiply by $\frac{x+3}{x+3}$ :

\frac{1}{x-3}\cdot\frac{x+3}{x+3} = \frac{x+3}{(x-3)(x+3)}.

Now subtract:

\frac{2x}{(x-3)(x+3)} - \frac{x+3}{(x-3)(x+3)}.

Step 4: Subtract the numerators.

Keep the common denominator and distribute the subtraction:

\frac{2x-(x+3)}{(x-3)(x+3)}.

Simplify the numerator:

2x-(x+3)=2x-x-3=x-3.

So,

\frac{x-3}{(x-3)(x+3)}.

Step 5: Simplify.

Cancel the common factor $x-3$ :

\boxed{\frac{1}{x+3}}, \qquad x\ne 3,-3.

Even though $x-3$ was canceled, $x=3$ is still excluded because it made an original denominator equal to zero. Never subtract denominators; only the numerators are subtracted after the fractions have a common denominator.

Learn by doing: Subtract rational expressions

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Rational Expressions - Add/Subtract Unlike Denominators (Polynomials), Final Answer

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