Simplifying Rational Expressions

Reducing rational expressions to simplest form

What is a Rational Expression?

A rational expression is a fraction with polynomials in the numerator and denominator.

Example: $\frac{x^2 - 4}{x + 2}$

Important: You can only cancel factors, not terms!

Values that make the denominator zero are excluded from the domain.

Example: $\frac{x + 3}{x - 5}$

Restriction: $x \neq 5$ (denominator would be zero)

❌ Wrong: $\frac{x + 3}{x} = 3$ (can't cancel terms!)

✓ Correct: $\frac{x + 3}{x}$ cannot be simplified further

Simplify: $\frac{6x^2}{3x}$

💡 Show Solution

Factor numerator and denominator: $\frac{6x^2}{3x} = \frac{3x \cdot 2x}{3x \cdot 1}$

Cancel the common factor $3x$ : $= \frac{2x}{1} = 2x$

Restriction: $x \neq 0$

Answer: $2x$ (where $x \neq 0$ )

Simplify: $\frac{x^2 - 9}{x^2 + 6x + 9}$

💡 Show Solution

Step 1: Factor the numerator (difference of squares) $x^2 - 9 = (x + 3)(x - 3)$

Step 2: Factor the denominator (perfect square trinomial) $x^2 + 6x + 9 = (x + 3)^2 = (x + 3)(x + 3)$

Step 3: Write and cancel $\frac{(x + 3)(x - 3)}{(x + 3)(x + 3)} = \frac{x - 3}{x + 3}$

Restriction: $x \neq -3$

Answer: $\frac{x - 3}{x + 3}$ (where $x \neq -3$ )

Simplify: $\frac{x^3 - 8}{x^2 - 4}$

💡 Show Solution

Step 1: Factor numerator (difference of cubes) $x^3 - 8 = x^3 - 2^3 = (x - 2)(x^2 + 2x + 4)$

Step 2: Factor denominator (difference of squares) $x^2 - 4 = (x + 2)(x - 2)$

Step 3: Write and cancel $(x - 2)$ $\frac{(x - 2)(x^2 + 2x + 4)}{(x + 2)(x - 2)} = \frac{x^2 + 2x + 4}{x + 2}$

Restrictions: $x \neq 2, -2$

Answer: $\frac{x^2 + 2x + 4}{x + 2}$ (where $x \neq \pm 2$ )

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