Algebraic fractions

1. Overview

Algebraic fractions are fractions where the numerator and/or denominator contain algebraic expressions. The key skill is to manipulate these fractions by simplifying, adding, subtracting, multiplying, and dividing them. This involves factorising, finding common denominators, and cancelling common factors. Proficiency with algebraic fractions is crucial for solving more advanced algebraic problems and is frequently tested in the IGCSE Mathematics exam (0580).

Key Definitions

• Rational Expression: An algebraic fraction where both the numerator and the denominator are polynomials.
• Simplifying: Reducing a fraction to its lowest terms by cancelling common factors from the numerator and denominator.
• Common Denominator: A shared multiple of the denominators of two or more fractions, required for addition and subtraction.
• Factorising: Breaking down an expression into a product of its factors (e.g., $x^2 - 4$ becomes $(x - 2)(x + 2)$).

Core Content

There are no specific "Core" objectives for this topic in the IGCSE syllabus; however, students should be comfortable with basic numerical fractions before attempting the Extended curriculum content below.

Extended Content (Extended Only)

A. Simplifying Algebraic Fractions

To simplify an algebraic fraction, you must factorise both the numerator and the denominator first. You can only cancel terms that are multiplied (factors), never terms that are added or subtracted.

Worked example 1 — Simplifying a rational expression

Question: Simplify the following expression: $\frac{2x^2 + 7x + 3}{x^2 + 5x + 6}$

1. Factorise the numerator:
   $2x^2 + 7x + 3 = (2x + 1)(x + 3)$
   Find two numbers that multiply to 6 (2 × 3) and add to 7 (1 and 6). Then rewrite as $2x^2 + x + 6x + 3$ and factorise by grouping.
2. Factorise the denominator:
   $x^2 + 5x + 6 = (x + 2)(x + 3)$
   Find two numbers that multiply to 6 and add to 5 (2 and 3).
3. Rewrite the fraction with factorised numerator and denominator:
   $\frac{(2x + 1)(x + 3)}{(x + 2)(x + 3)}$
4. Cancel the common factor:
   $\frac{(2x + 1)\cancel{(x + 3)}}{(x + 2)\cancel{(x + 3)}}$
   $(x+3)$ appears in both numerator and denominator.
5. Final simplified form:
   $\frac{2x + 1}{x + 2}$

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B. Multiplication and Division

• Multiplication: Multiply the numerators together and the denominators together. Simplify by cancelling factors before multiplying to make it easier.
• Division: "Keep, Change, Flip." Keep the first fraction, change the sign to multiplication, and flip the second fraction (the reciprocal).

Worked example 2 — Dividing algebraic fractions

Question: Simplify $\frac{x^2 - 4}{x^2 + x} \div \frac{x - 2}{x}$

1. Rewrite as multiplication by the reciprocal:
   $\frac{x^2 - 4}{x^2 + x} \times \frac{x}{x - 2}$
   "Keep, Change, Flip" - invert the second fraction and change division to multiplication.
2. Factorise the numerator of the first fraction:
   $x^2 - 4 = (x - 2)(x + 2)$
   Difference of two squares.
3. Factorise the denominator of the first fraction:
   $x^2 + x = x(x + 1)$
4. Rewrite the expression with factorised terms:
   $\frac{(x - 2)(x + 2)}{x(x + 1)} \times \frac{x}{x - 2}$
5. Cancel common factors:
   $\frac{\cancel{(x - 2)}(x + 2)}{x(x + 1)} \times \frac{x}{\cancel{(x - 2)}}$
6. Cancel common factors:
   $\frac{(x + 2)}{\cancel{x}(x + 1)} \times \frac{\cancel{x}}{1}$
7. Final Answer:
   $\frac{x + 2}{x + 1}$

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C. Addition and Subtraction

To add or subtract, you must find a Common Denominator. The easiest way is to multiply the two denominators together.

Worked example 3 — Adding algebraic fractions

Question: Express as a single fraction: $\frac{4}{x - 3} - \frac{1}{x + 2}$

1. Find a common denominator:
   $(x - 3)(x + 2)$
2. Adjust the numerators:
   $\frac{4(x + 2)}{(x - 3)(x + 2)} - \frac{1(x - 3)}{(x - 3)(x + 2)}$
3. Expand the numerators:
   $\frac{4x + 8 - (x - 3)}{(x - 3)(x + 2)}$
4. Simplify the numerator, being careful with the negative sign:
   $\frac{4x + 8 - x + 3}{(x - 3)(x + 2)}$
   Remember to distribute the negative sign to both terms in the second numerator.
5. Collect like terms in the numerator:
   $\frac{3x + 11}{(x - 3)(x + 2)}$
6. Final Answer:
   $\frac{3x + 11}{(x - 3)(x + 2)}$

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Worked example 4 — Combining fractions with more complex denominators

Question: Express as a single fraction in its simplest form: $\frac{5}{2x + 1} + \frac{2}{4x^2 - 1}$

1. Factorise the second denominator:
   $4x^2 - 1 = (2x - 1)(2x + 1)$
   This is a difference of two squares.
2. Identify the common denominator:
   $(2x - 1)(2x + 1)$
   Notice that $(2x+1)$ is already a factor of the second denominator.
3. Adjust the first fraction:
   $\frac{5(2x - 1)}{(2x + 1)(2x - 1)} + \frac{2}{(2x - 1)(2x + 1)}$
4. Expand the numerator of the first fraction:
   $\frac{10x - 5}{(2x + 1)(2x - 1)} + \frac{2}{(2x - 1)(2x + 1)}$
5. Combine the numerators:
   $\frac{10x - 5 + 2}{(2x + 1)(2x - 1)}$
6. Simplify the numerator:
   $\frac{10x - 3}{(2x + 1)(2x - 1)}$
7. Final Answer:
   $\frac{10x - 3}{(2x + 1)(2x - 1)}$

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Key Equations

Operation	General Formula	Notes
Multiplication	$\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$	Cancel common factors before multiplying.
Division	$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$	"Flip" the second fraction only.
Addition	$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$	Find the Lowest Common Multiple (LCM) of $b$ and $d$ for easier simplification.
Subtraction	$\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}$	Find the Lowest Common Multiple (LCM) of $b$ and $d$ for easier simplification.

Common Mistakes to Avoid

• ❌ Wrong (Incorrect Cancellation): $\frac{x^2 + 4x}{x} = x^2 + 4$
• ✅ Right: You must factorise first: $\frac{x(x + 4)}{x} = x + 4$. You can only cancel factors, not terms.
• ❌ Wrong (Sign Error in Subtraction): $\frac{5}{x} - \frac{x + 2}{3} = \frac{15 - x + 2}{3x}$
• ✅ Right: Use brackets to ensure the negative sign applies to the entire numerator: $\frac{5(3) - x(x + 2)}{3x} = \frac{15 - x^2 - 2x}{3x}$.
• ❌ Wrong (Incomplete Factorisation): Simplifying $\frac{x^2 - 1}{x + 1}$ to $x - 1$ without showing the factorisation $(x-1)(x+1)$ in your working.
• ✅ Right: Show the factorisation: $\frac{(x - 1)(x + 1)}{(x + 1)} = x - 1$. This demonstrates your understanding.

Exam Tips

• Command Words: Look for "Simplify fully" or "Express as a single fraction in its simplest form." The phrase "simplest form" always implies factorisation and cancellation.
• Show Your Working: Even if you make a small arithmetic error, you can still earn method marks for correctly factorising, finding a common denominator, or applying the "keep, change, flip" rule.
• Difference of Two Squares & Quadratics: Be on the lookout for expressions like $x^2 - 9$ or $4x^2 - 1$ (difference of two squares) and general quadratic expressions that can be factorised. These are very common in algebraic fraction questions.
• Calculator Use: While calculators can help with arithmetic, they won't factorise algebraic expressions for you. Focus on mastering the algebraic manipulation techniques.
• Final Form: Unless the question specifically asks you to expand the denominator, leaving it in factorised form (e.g., $(x+1)(x-2)$) is perfectly acceptable and often preferred. This makes it easier for the examiner to check your work.
• Double-Check: After simplifying, quickly check if your answer makes sense by substituting a simple value for $x$ (e.g., $x = 1$) into both the original expression and your simplified answer. If the results are different, you've made a mistake.