Course Notes Practice Past Paper Archive

Algebraic fractions0%

Algebra · Topic 7 of 12

Algebraic fractions

Video lesson7 worked examples

Theory

Simplifying

You can only simplify an algebraic fraction if there is a common factor or identical bracket on the top and bottom. You cannot cancel individual terms inside brackets; you must fully factorise first.

Four Operations

You must be able to add, subtract, multiply, and divide algebraic fractions, expressing them in their simplest form. To add or subtract, use a common denominator; to divide, flip the second fraction and multiply.

The Golden Rule: you can only cancel factors, never individual terms — so always factorise fully first, then cancel identical brackets. For adding and subtracting, the common denominator of two different brackets is simply their product.

⚠️ Common Examiner Traps

Cancelling terms, not factors: in $\frac{x+2}{x+4}$ you cannot cancel the $x$ s or the numbers — nothing cancels unless it is a whole bracket top and bottom.
Subtraction sign: when subtracting, the minus applies to the whole numerator of the second fraction — bracket it, e.g. $\ldots - (3x - 1)$ .
Dividing: “keep, change, flip” — flip the second fraction and multiply. Do not flip the first.
Stopping too early: after combining or multiplying, check whether the result factorises and cancels further.

Worked examples

Example 1

Simplifying (difference of squares)

Simplify $\dfrac{(x - 4)(x + 2)}{x^2 - 16}$ .

Step 1: Factorise the bottom using the difference of two squares: $\dfrac{(x - 4)(x + 2)}{(x - 4)(x + 4)}$ .

Step 2: Cancel the identical $(x - 4)$ brackets. Answer: $\dfrac{x + 2}{x + 4}$ .

Example 2

Simplifying (factorising a trinomial)

Simplify $\dfrac{x^2 + 5x + 6}{x^2 - 9}$ .

Step 1: Factorise top and bottom fully. The top is a trinomial, the bottom a difference of squares:

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

Step 2: Cancel the common $(x + 3)$ bracket:

Answer: $\dfrac{x + 2}{x - 3}$ .

Example 3

Adding (numeric denominators)

Express $\dfrac{a}{3} + \dfrac{b}{4}$ as a single fraction.

Step 1: Find a common denominator (12) and scale the numerators: $\dfrac{4a}{12} + \dfrac{3b}{12}$ .

Answer: $\dfrac{4a + 3b}{12}$ .

Example 4

Subtracting (algebraic denominators)

Express $\dfrac{4}{x - 1} - \dfrac{3}{x + 2}$ as a single fraction $(x \neq 1,\ x \neq -2)$ .

Step 1: The common denominator is the product of the two brackets, $(x - 1)(x + 2)$ . Scale each numerator by the other bracket:

\dfrac{4(x + 2) - 3(x - 1)}{(x - 1)(x + 2)}

Step 2: Expand the numerator carefully — the minus applies to all of $3(x-1)$ :

4x + 8 - 3x + 3 = x + 11

Answer: $\dfrac{x + 11}{(x - 1)(x + 2)}$ .

Example 5

Multiplying

Express $\dfrac{5}{2a} \times \dfrac{a^2}{10}$ in its simplest form.

Step 1: Multiply straight across the tops and bottoms: $\dfrac{5 \times a^2}{2a \times 10} = \dfrac{5a^2}{20a}$ .

Step 2: Simplify — divide numbers by 5 and cancel one $a$ : $\dfrac{5a^2}{20a} = \dfrac{a}{4}$ .

Answer: $\dfrac{a}{4}$ .

Example 6

Dividing

Express $\dfrac{2}{3x} \div \dfrac{4}{9x^2}$ in its simplest form.

Step 1: Keep, change, flip — flip the second fraction and multiply: $\dfrac{2}{3x} \times \dfrac{9x^2}{4}$ .

Step 2: Multiply across: $\dfrac{18x^2}{12x}$ .

Step 3: Simplify — divide numbers by 6 and cancel one $x$ : $\dfrac{18x^2}{12x} = \dfrac{3x}{2}$ .

Answer: $\dfrac{3x}{2}$ .

Example 7

🔗 Bringing it together

Simplify $\dfrac{x^2 - 4x}{x^2 - x - 12}$ .

Step 1: Nothing cancels yet, so factorise both parts fully. The top has a common factor; the bottom is a trinomial:

\dfrac{x(x - 4)}{(x - 4)(x + 3)}

Step 2: Cancel the common $(x - 4)$ bracket:

Answer: $\dfrac{x}{x + 3}$ .

Ready to practise?

22 Algebraic Fractions questions with answers and video solutions