Working with Algebraic Fractions

1. Simplifying

The key is to factorise the numerator and denominator, then cancel any common factors.

Example:
$\frac{x^2 + 7x + 10}{x + 2} = \frac{(x+5)(x+2)}{x+2} = x + 5$

⚠️ You can only cancel factors, not terms being added or subtracted!

2. Multiplying

Multiply numerators together and denominators together. It’s often easier to factorise and cancel first.

Example:
$\frac{2x+4}{x^2} \times \frac{x}{x+2} = \frac{2(x+2)}{x \cdot x} \times \frac{x}{x+2} = \frac{2}{x}$

3. Dividing

Use the Keep, Change, Flip (KCF) rule: keep the first fraction, change divide (÷) to multiply (×), and flip the second fraction upside down. Then multiply as normal.

Rule:
$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$

4. Adding & Subtracting

You must have a common denominator — usually the product of the two denominators.

Example:
$\frac{2}{x+1} – \frac{3}{x-2}$
Common denominator: $(x+1)(x-2)$
$= \frac{2(x-2) – 3(x+1)}{(x+1)(x-2)}$
$= \frac{2x-4-3x-3}{(x+1)(x-2)}$
$= \frac{-x-7}{(x+1)(x-2)}$

Simplifying

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

1. Factorise top and bottom:
   Top (difference of squares): $x^2 – 4 = (x-2)(x+2)$
   Bottom (quadratic): $x^2 + x – 2 = (x+2)(x-1)$
2. Rewrite the fraction:
   $\frac{(x-2)(x+2)}{(x+2)(x-1)}$
3. Cancel the $(x+2)$ factor.

Answer: $\frac{x-2}{x-1}$

Dividing

Simplify $\frac{m^2 – 25}{m+1} \div \frac{m-5}{m^2 + 2m + 1}$.

1. Keep, Change, Flip:
   $\frac{m^2-25}{m+1} \times \frac{m^2+2m+1}{m-5}$
2. Factorise everything:
   $\frac{(m-5)(m+5)}{(m+1)} \times \frac{(m+1)^2}{(m-5)}$
3. Cancel $(m-5)$ and one $(m+1)$:
   $= (m+5)(m+1)$

Answer: $(m+5)(m+1)$, or equivalently $m^2 + 6m + 5$.

🤔 Common Misunderstandings

• Cancelling terms, not factors: The biggest mistake! You cannot cancel the $x$ in $\frac{x+1}{x+2}$. You must factorise first.
• Sign errors in subtraction: Forgetting to apply the minus sign to all terms in the second numerator. Always use brackets around the numerator you’re subtracting.

📝 Common Exam Mistakes

• Weak factorising skills: If you struggle with factorising quadratics, you’ll struggle with algebraic fractions. It’s the most important building block.
• Incorrect LCM: Choosing the wrong common denominator for addition/subtraction.
• Not simplifying the final answer after performing an operation.