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Algebraic Fractions

Fractions with expressions on top and bottom follow exactly the same rules as ordinary number fractions. The single new skill is factorising first, so you can see what cancels.

MYP 4Extended MathsAlgebraCriteria A · C · D~10 min read

Written by Fayek · MYP student

Simplifying by factorising and cancelling

An algebraic fraction (also called a rational expression) is just a fraction whose numerator and denominator are algebraic expressions. To simplify one, you factorise the top and the bottom fully, then cancel any factor that appears in both.

The golden rule of cancelling
You may only cancel factors, never individual terms. A factor is something multiplied across the whole line; a term is something added. In \(\dfrac{(x-3)(x+3)}{(x+3)(x+4)}\) the bracket \((x+3)\) is a factor, so it cancels. You could never cancel the loose \(x\) values in \(\dfrac{x+3}{x+4}\).
Worked example

Simplify \(\dfrac{x^2 - 9}{x^2 + 7x + 12}\).

1
Factorise the numerator as a difference of two squares: \(x^2 - 9 = (x - 3)(x + 3)\).
2
Factorise the denominator as a trinomial: \(x^2 + 7x + 12 = (x + 3)(x + 4)\).
3
Rewrite the fraction: \(\dfrac{(x - 3)(x + 3)}{(x + 3)(x + 4)}\).
4
Cancel the common factor \((x + 3)\).
\(\dfrac{x - 3}{x + 4}\)

Where this is assessed

This topic lives in Criterion A (applying factorising and fraction rules) and rewards Criterion C heavily: examiners want to see each factorising line and each cancelling step written clearly, not just a final fraction.

Multiplying and dividing

Multiplying rational expressions is the friendliest operation: factorise everything, cancel across the top and bottom of the whole product, then write what is left. There is no need for a common denominator.

Worked example

Simplify \(\dfrac{x + 2}{x - 3} \times \dfrac{x^2 - 9}{x^2 + 4x + 4}\).

1
Factorise every part: \(x^2 - 9 = (x - 3)(x + 3)\) and \(x^2 + 4x + 4 = (x + 2)^2\).
2
Write it all as one product: \(\dfrac{(x + 2)\,(x - 3)(x + 3)}{(x - 3)\,(x + 2)(x + 2)}\).
3
Cancel the shared \((x - 3)\) and one \((x + 2)\).
\(\dfrac{x + 3}{x + 2}\)

Dividing is one step longer: flip the second fraction and multiply. Never try to cancel until you have turned the division into a multiplication.

Worked example

Simplify \(\dfrac{x^2 - 4}{x + 5} \div \dfrac{x - 2}{2x + 10}\).

1
Flip the second fraction and change to multiply: \(\dfrac{x^2 - 4}{x + 5} \times \dfrac{2x + 10}{x - 2}\).
2
Factorise: \(x^2 - 4 = (x - 2)(x + 2)\) and \(2x + 10 = 2(x + 5)\).
3
Write as one product: \(\dfrac{(x - 2)(x + 2)\,\cdot\,2(x + 5)}{(x + 5)\,(x - 2)}\).
4
Cancel \((x + 5)\) and \((x - 2)\).
\(2(x + 2)\)

Adding and subtracting

Adding and subtracting need a common denominator, exactly as with number fractions. The safe common denominator is the product of the two bottoms (or their lowest common multiple if they share a factor). Rewrite each fraction over that denominator, then combine the numerators. Take extra care with subtraction: the minus sign applies to every term of the second numerator.

Worked example

Write \(\dfrac{2}{x} + \dfrac{3}{x + 1}\) as a single fraction.

1
The common denominator is \(x(x + 1)\).
2
Rewrite each fraction over it: \(\dfrac{2(x + 1)}{x(x + 1)} + \dfrac{3x}{x(x + 1)}\).
3
Add the numerators: \(2(x + 1) + 3x = 2x + 2 + 3x = 5x + 2\).
\(\dfrac{5x + 2}{x(x + 1)}\)

The subtraction trap

For \(\dfrac{3}{x - 2} - \dfrac{1}{x + 1}\), the common denominator is \((x - 2)(x + 1)\), giving \(\dfrac{3(x + 1) - 1(x - 2)}{(x - 2)(x + 1)}\). The top becomes \(3x + 3 - x + 2 = 2x + 5\), because subtracting \(-2\) turns into \(+2\). Forgetting to distribute that minus over both terms is the single most common slip. The answer is \(\dfrac{2x + 5}{(x - 2)(x + 1)}\).

Factorise before you commit

Always factorise every denominator first. If they share a factor, the lowest common denominator is smaller than the product, and your numbers stay tidy. Then check the final numerator: if it factorises and cancels with the denominator, simplify once more.

Check yourself

Try each with the answers hidden. Show your factorising.

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

Factorise top and bottom: \(x^2 - 1 = (x - 1)(x + 1)\) and \(x^2 - x = x(x - 1)\). Cancel \((x - 1)\) to get \(\dfrac{x + 1}{x}\).

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

Factorise \(x^2 - 4 = (x - 2)(x + 2)\), so the product is \(\dfrac{x\,(x - 2)(x + 2)}{(x + 2)\,x^2}\). Cancel \((x + 2)\) and one \(x\): \(\dfrac{x - 2}{x}\).

3. Write \(\dfrac{1}{x} + \dfrac{1}{x + 3}\) as a single fraction. +

Common denominator \(x(x + 3)\): \(\dfrac{(x + 3) + x}{x(x + 3)} = \dfrac{2x + 3}{x(x + 3)}\). The answer is \(\dfrac{2x + 3}{x(x + 3)}\).

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