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Operating with
Rational Expressions

Master the four operations: add, subtract, multiply, and divide polynomial fractions, just like you do with regular fractions.

Algebra 2

What Is a Rational Expression?

A rational expression is any expression that can be written as the ratio of two polynomials. Think of it as a fraction where the numerator and denominator are polynomials instead of plain numbers.

$$ \frac{P(x)}{Q(x)} \quad \text{where } Q(x) \neq 0 $$
Rational Number
\(\dfrac{3}{5}\) : ratio of two integers
Rational Expression
\(\dfrac{x^2 + 1}{x - 3}\) : ratio of two polynomials

The Closure Property

Rational expressions are closed under addition, subtraction, multiplication, and division (by a nonzero expression). This means performing any of these operations on two rational expressions always yields another rational expression.

Why Does This Matter?

Simplify Models
Physics and engineering formulas often combine rational expressions (e.g., electrical resistance in parallel).
Solve Equations
Adding or subtracting rational expressions is essential for solving rational equations.
Build Toward Calculus
Partial fraction decomposition, limits, and integration all require fluency with these operations.

The Four Operations at a Glance

Multiply
\(\dfrac{A}{B} \cdot \dfrac{C}{D} = \dfrac{A \cdot C}{B \cdot D}\)
Divide
\(\dfrac{A}{B} \div \dfrac{C}{D} = \dfrac{A}{B} \cdot \dfrac{D}{C}\)
Add
\(\dfrac{A}{B} + \dfrac{C}{D} = \dfrac{AD + BC}{BD}\)
Subtract
\(\dfrac{A}{B} - \dfrac{C}{D} = \dfrac{AD - BC}{BD}\)

Multiplying Rational Expressions

Multiplication is the simplest operation: factor everything, cancel common factors, then multiply across.

$$ \frac{A}{B} \cdot \frac{C}{D} = \frac{A \cdot C}{B \cdot D} $$

Dividing: Multiply by the Reciprocal

To divide, flip the second fraction (take its reciprocal) and then multiply. The divisor must be nonzero.

$$ \frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \cdot \frac{D}{C}, \quad C \neq 0 $$

Adding & Subtracting: Finding the LCD

Just like numeric fractions, you need a common denominator before you can add or subtract rational expressions. The Least Common Denominator (LCD) is the simplest expression that both denominators divide into.

1
Factor each denominator completely.
2
List every distinct factor at its highest power.
3
Multiply those factors together to get the LCD.

Using the LCD vs. A Common Denominator

Use the LCD Just Multiply Denominators
Result size Simplest possible denominator Often unnecessarily large
Extra simplification Minimal or none Usually required at the end
When to use Always preferred Quick trick when denominators share no factors

How to Add or Subtract

1
Factor every denominator.
2
Find the LCD from all distinct factors.
3
Rewrite each fraction with the LCD as its denominator.
4
Combine the numerators (add or subtract).
5
Simplify: factor the resulting numerator and cancel any common factors with the denominator.

Example: Multiplying

Simplify: \(\dfrac{x^2 - 9}{2x} \cdot \dfrac{4x^2}{x + 3}\)

1
Factor: \(x^2 - 9 = (x-3)(x+3)\)
2
Rewrite: \(\dfrac{(x-3)(x+3)}{2x} \cdot \dfrac{4x^2}{x+3}\)
3
Cancel \((x+3)\) and simplify \(\dfrac{4x^2}{2x} = 2x\)
4
Result: \(\boxed{2x(x-3)}\)

Example: Dividing

Simplify: \(\dfrac{x^2 - 4}{x^2 + 5x + 6} \div \dfrac{x - 2}{x + 1}\)

1
Flip: \(\dfrac{x^2 - 4}{x^2 + 5x + 6} \cdot \dfrac{x + 1}{x - 2}\)
2
Factor: \(\dfrac{(x-2)(x+2)}{(x+2)(x+3)} \cdot \dfrac{x+1}{x-2}\)
3
Cancel \((x-2)\) and \((x+2)\)
4
Result: \(\boxed{\dfrac{x+1}{x+3}}\)

Example: Adding

Simplify: \(\dfrac{3}{x+2} + \dfrac{5}{x-1}\)

1
LCD: \((x+2)(x-1)\)
2
Rewrite: \(\dfrac{3(x-1)}{(x+2)(x-1)} + \dfrac{5(x+2)}{(x+2)(x-1)}\)
3
Combine: \(\dfrac{3x - 3 + 5x + 10}{(x+2)(x-1)}\)
4
Result: \(\boxed{\dfrac{8x + 7}{(x+2)(x-1)}}\)

Example: Subtracting with Shared Factors

Simplify: \(\dfrac{x}{x^2 - 1} - \dfrac{2}{x + 1}\)

1
Factor: \(x^2 - 1 = (x-1)(x+1)\), so LCD = \((x-1)(x+1)\)
2
Rewrite: \(\dfrac{x}{(x-1)(x+1)} - \dfrac{2(x-1)}{(x-1)(x+1)}\)
3
Combine: \(\dfrac{x - 2x + 2}{(x-1)(x+1)} = \dfrac{-x + 2}{(x-1)(x+1)}\)
4
Result: \(\boxed{\dfrac{2-x}{(x-1)(x+1)}}\)   or equivalently \(\dfrac{-(x-2)}{(x-1)(x+1)}\)

Common Mistakes to Avoid

✕ Cancelling Terms
Cancelling \(x\) in \(\dfrac{x + 3}{x}\) to get \(3\). You can only cancel factors, not terms being added.
✕ Forgetting to Distribute the Negative
In subtraction, forgetting that the minus sign applies to every term in the second numerator: \(-(2x - 1) = -2x + 1\).
✕ Skipping Factoring
Multiplying denominators without factoring first. This can create unnecessarily complex expressions that are hard to simplify.

Summary of Operations

Operation Formula Key Step
Multiply \(\dfrac{A}{B} \cdot \dfrac{C}{D} = \dfrac{AC}{BD}\) Factor, then cancel
Divide \(\dfrac{A}{B} \div \dfrac{C}{D} = \dfrac{A}{B} \cdot \dfrac{D}{C}\) Flip, then multiply
Add \(\dfrac{A}{B} + \dfrac{C}{D} = \dfrac{AD+BC}{BD}\) Find LCD first
Subtract \(\dfrac{A}{B} - \dfrac{C}{D} = \dfrac{AD-BC}{BD}\) Distribute the negative

Question 1 of 5

Simplify: \(\dfrac{x^2 - 25}{x + 5} \cdot \dfrac{x}{x - 5}\)

Question 2 of 5

Simplify: \(\dfrac{2}{x+3} + \dfrac{x}{x-2}\)

Question 3 of 5

Simplify: \(\dfrac{x^2 + 2x}{x^2 - 4} \div \dfrac{x}{x - 2}\)

Question 4 of 5

What is the LCD of \(\dfrac{3}{x^2 - 1}\) and \(\dfrac{7}{x^2 + x}\)?

Question 5 of 5

Which expression equals \(\dfrac{5}{x-4} - \dfrac{3}{x+2}\)?