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Lesson overview

Algebra - rational expressions

Algebraic fractions follow the same rules as numerical fractions, but the numerators and denominators are algebraic expressions. To simplify an algebraic fraction, factor both the numerator and denominator, then cancel common factors. When adding or subtracting algebraic fractions, find a common denominator (often the product of the two denominators, or their LCM). When multiplying, multiply numerators together and denominators together, then simplify. When dividing, multiply by the reciprocal of the divisor. These skills are essential for solving equations involving fractions and for calculus work.

Simplifying by cancelling common factors

Simplify $\frac{x^2 + 5x}{x}$ .

Factor the numerator. The common factor $x$ cancels from numerator and denominator.

Simplifying with quadratic factors

Simplify $\frac{x^2 - 4}{x - 2}$ .

Recognise $x^2 - 4$ as a difference of squares. Factor and cancel.

Multiplying algebraic fractions

Simplify $\frac{3x}{2y} \times \frac{4y}{x^2}$ .

Multiply numerators and denominators. Cancel common factors: $x$ and $y$ .

Dividing algebraic fractions

Simplify $\frac{2x}{3} \div \frac{x}{6}$ .

Flip the second fraction and multiply. Then simplify by cancelling.

Adding with a common denominator

Simplify $\frac{2}{x} + \frac{3}{x}$ .

When denominators are the same, add the numerators and keep the denominator.

Worked example: Adding and subtracting algebraic fractions

Simplify $\frac{3}{x + 2} - \frac{2}{x - 1}$ .

The denominators are $(x + 2)$ and $(x - 1)$ . These don't share common factors, so the LCD is $(x + 2)(x - 1)$ .
Rewrite the first fraction with denominator $(x + 2)(x - 1)$ : multiply by $\frac{x - 1}{x - 1}$ to get $\frac{3(x - 1)}{(x + 2)(x - 1)}$ .
Rewrite the second fraction: multiply by $\frac{x + 2}{x + 2}$ to get $\frac{2(x + 2)}{(x + 2)(x - 1)}$ .
Subtract the numerators: $\frac{3(x - 1) - 2(x + 2)}{(x + 2)(x - 1)}$ .
Expand: $3(x - 1) = 3x - 3$ and $2(x + 2) = 2x + 4$ .
Numerator: $3x - 3 - 2x - 4 = x - 7$ .
Final answer: $\frac{x - 7}{(x + 2)(x - 1)}$ .

Try it

Simplify each algebraic fraction. Cancel common factors, find common denominators, or apply multiplication and division rules. Choose the correct simplified form.

Question 1

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

💡 Cancel the common factor of $3x$ from numerator and denominator.

Question 2

Simplify $\frac{x^2 - 9}{x - 3}$ .

💡 $x^2 - 9 = (x - 3)(x + 3)$ . Cancel the $(x - 3)$ factor.

Question 3

Multiply: $\frac{4}{x} \times \frac{x^2}{8}$ .

💡 $\frac{4x^2}{8x} = \frac{4x}{8} = \frac{x}{2}$ .

Question 4

Divide: $\frac{5}{2x} \div \frac{10}{x}$ .

💡 $\frac{5}{2x} \times \frac{x}{10} = \frac{5x}{20x} = \frac{1}{4}$ .

Question 5

Add: $\frac{2}{x} + \frac{1}{x + 1}$ .

💡 LCD is $x(x + 1)$ . Rewrite: $\frac{2(x + 1)}{x(x + 1)} + \frac{x}{x(x + 1)} = \frac{2x + 2 + x}{x(x + 1)} = \frac{3x + 2}{x(x + 1)}$ .

Common mistakes

Watch for these when working through the lesson.

Cancelling terms instead of factors. You can only cancel common factors that appear in both numerator and denominator, not terms that are added or subtracted.
Forgetting to multiply both numerator and denominator by the same expression when finding a common denominator. This changes the fraction's value.
When dividing, forgetting to flip the second fraction. Division by a fraction is multiplication by its reciprocal.

Algebraic fractions

Lesson overview

Worked example: Adding and subtracting algebraic fractions

Try it

Common mistakes

Related topics

Independent practice will plug in here