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Lesson overview

Algebra — algebraic fractions

Algebraic fractions follow the same rules as numerical fractions. To simplify: factorise the numerator and denominator, then cancel common factors. For example, $\frac{x^2 - 1}{x + 1} = \frac{(x - 1)(x + 1)}{x + 1} = x - 1$ (provided $x \neq -1$ ). To add or subtract algebraic fractions, find a common denominator. To multiply: $\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$ . To divide: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$ . Always state any restrictions on variables (e.g., the denominator must not be zero).

Simplifying algebraic fractions by factorising and cancelling

Simplify $\frac{6x + 12}{3x}$ .

Factorise the numerator, then cancel common factors. Here, 6 and 3 share a common factor of 3.

Simplifying by cancelling quadratic factors

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

Recognise the difference of two squares in the numerator. Factor and cancel, noting the restriction on x.

Adding algebraic fractions with different denominators

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

Find the common denominator, multiply each fraction's numerator and denominator appropriately, then combine.

Multiplying algebraic fractions

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

Multiply numerators and denominators. Then factorise and cancel. Note any restrictions.

Worked example — combining and simplifying

Simplify $\frac{2x}{x + 1} - \frac{x}{x}$ . (Note: assume $x \neq 0$ and $x \neq -1$ .)

Simplify any fractions that can be reduced: $\frac{x}{x} = 1$ (where $x \neq 0$ ).
Rewrite the expression as $\frac{2x}{x + 1} - 1$ .
Express 1 with denominator $(x + 1)$ : $\frac{2x}{x + 1} - \frac{x + 1}{x + 1}$ .
Subtract the numerators: $\frac{2x - (x + 1)}{x + 1}$ .
Simplify the numerator: $2x - x - 1 = x - 1$ .
The final answer is $\frac{x - 1}{x + 1}$ , where $x \neq 0$ and $x \neq -1$ .

Try it

Work carefully with factorisation and common denominators. Always state restrictions on variables and check your simplifications.

Question 1

Simplify $\frac{4x}{2x + 2}$ .

💡 Factorise the denominator: $2x + 2 = 2(x + 1)$ . Cancel 2.

Question 2

Simplify $\frac{x^2 - 1}{x - 1}$ .

💡 Factorise: $x^2 - 1 = (x - 1)(x + 1)$ . Cancel $(x - 1)$ .

Question 3

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

💡 Common denominator is $2x$ . First fraction becomes $\frac{2}{2x}$ , second becomes $\frac{x}{2x}$ .

Question 4

Simplify $\frac{3(x - 1)}{x - 1} \times \frac{2}{x}$ .

💡 Cancel $(x - 1)$ from the first fraction. Then multiply: $3 \times \frac{2}{x}$ .

Question 5

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

💡 Factorise the numerator: $x^2 + 3x = x(x + 3)$ . Then divide by $x$ .

Common mistakes

Watch for these when working through the lesson.

Cancelling incorrectly. You can only cancel factors, not terms. $\frac{x + 2}{x}$ does NOT simplify to $\frac{2}{1}$ .
Forgetting to state restrictions. If you cancel $(x - 1)$ , you must note that $x \neq 1$ .
When combining fractions, forgetting to multiply the entire numerator by the scale factor, not just one term.
Not fully factorising before cancelling. Always check for common factors in the numerator and denominator.

Manipulate algebraic fractions

Lesson overview

Worked example — combining and simplifying

Try it

Common mistakes

Related topics

Independent practice will plug in here