1. Simplifying Algebraic Fractions
What is an Algebraic Fraction?
An algebraic fraction is simply a fraction where the numerator (top) and/or denominator (bottom) contain algebraic expressions instead of just numbers.
For example:
$\frac{2x}{x^2 + 3x}$ or $\frac{x + 5}{2x - 1}$
Think of it this way: if you know how to simplify numerical fractions like $\frac{6}{9} = \frac{2}{3}$, algebraic fractions work the exact same way. But instead of finding the GCF of numbers, you'll find common factors in expressions.
The Golden Rule: Factorise First
To simplify any algebraic fraction, follow this pattern:
STEP 1: Factorise the numerator fully. STEP 2: Factorise the denominator fully. STEP 3: Cancel any common factors that appear on both top and bottom.
The key insight: you can only cancel factors, not terms. If something is added or subtracted (a term), it's not available for cancelling until you factor it out.
Example: Simplifying with a Single Term Factor
Simplify: $\frac{x(5x - 1)}{4x}$
Step 1 & 2: Both top and bottom are already factored.
Step 3: The factor $x$ appears on both top and bottom, so cancel it:
$\frac{\cancel{x}(5x - 1)}{4\cancel{x}} = \frac{5x - 1}{4}$
This works because $x ÷ x = 1$, which disappears.
Example: Simplifying with a Bracket Factor
Sometimes the common factor is an entire bracket:
Simplify: $\frac{x(x + 2)}{(x + 2)(x - 1)}$
Both the top and bottom have $(x + 2)$ as a factor. Cancel it:
$\frac{x\cancel{(x + 2)}}{\cancel{(x + 2)}(x - 1)} = \frac{x}{x - 1}$
A Critical Mistake: Not Everything Can Be Cancelled
❌ Wrong: Trying to simplify $\frac{6x}{x + 1}$ by "cancelling the $x$"
Why it's wrong: The $x$ in the denominator is part of a sum ($x + 1$). It's a term, not a factor. You cannot factor out $x$ from $x + 1$ because the $+1$ doesn't contain an $x$.
✓ Correct answer: The fraction $\frac{6x}{x + 1}$ is already in simplest form.
Worked Example: A Complete Simplification
Simplify: $\frac{4x + 6}{2x^2 - 7x - 15}$
Step 1: Factorise the numerator
$4x + 6$ — both terms are even, so factor out 2:
$4x + 6 = 2(2x + 3)$
Step 2: Factorise the denominator
$2x^2 - 7x - 15$ — use the "pair method" or factoring by grouping. We need two numbers that multiply to $2 × (-15) = -30$ and add to $-7$. Those are $-10$ and $3$.
$2x^2 - 10x + 3x - 15 = 2x(x - 5) + 3(x - 5) = (2x + 3)(x - 5)$
Step 3: Cancel common factors
$\frac{2(2x + 3)}{(2x + 3)(x - 5)} = \frac{2}{x - 5}$
Notice: the hint was in the numerator's factor $(2x + 3)$. This told us to look for $(2x + 3)$ in the denominator!
⭐ Exam Tips
- Always factorise top and bottom first before trying to simplify
- If the top and bottom share a common factor, you'll almost always find it this way
- If one expression is hard to factorise, the other one might give you a hint about what to look for
- Check your final answer: can it be factored further? If not, you're done