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Lesson 3

Simplifying Algebraic Fractions by Factorisation

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Hello! Welcome to our next lesson.

In the past few sessions, we've focused on building a solid foundation in factorising—first with general quadratics, then with the special case of the difference of two squares. Today, we'll put those skills to immediate and practical use. Our goal is to simplify algebraic fractions by factorising the numerator and denominator.

Think of this as the algebraic equivalent of code refactoring. Just as a programmer simplifies complex code to make it more efficient and readable, we will simplify complex fractions to make them easier to work with. This is a vital skill for solving the more advanced equations you'll encounter later, particularly when analysing functions in game development or AI.

The core idea is simple: if you can factorise the top and bottom of a fraction, you can often find common factors to cancel out, leaving you with a much tidier expression. Let's begin.

Part 1: A Quick Factorising Refresher

Since this entire topic depends on your ability to factorise quickly and accurately, let's start with a quick review. The following video gives a great summary of the three main types of factorising you'll need for this lesson. It will be a good chance to solidify your knowledge, especially around quadratic factorisation, which you mentioned was an area you wanted to strengthen.

Algebraic Fractions (Simplifying) - GCSE Higher Maths

This video from 1st Class Maths provides a concise overview of the key factorising techniques required for simplifying algebraic fractions.

Please watch the introductory section, from the beginning up to the main examples. Pay close attention to the recap of single bracket factorisation, the review of double bracket factorisation for quadratics, and the section on the difference of two squares. This covers everything from our previous lessons that we'll need today.

Part 2: The Core Method - Factorise and Cancel

The principle behind simplifying algebraic fractions is identical to simplifying numerical fractions. To simplify 20/28, you would find the highest common factor (4), rewrite it as (4 * 5) / (4 * 7), and then cancel the common factor of 4 to get 5/7.

We do exactly the same with algebra. The 'rewriting' step is factorisation.

Following your preferred approach, let's try a problem first. Have a go at simplifying this fraction:

(x^2 - 3x - 10) / (x^2 + 2x - 35)

Take a moment to factorise the numerator and the denominator. What do you get?

...

You should find that:

  • The numerator x^2 - 3x - 10 factorises to (x - 5)(x + 2).
  • The denominator x^2 + 2x - 35 factorises to (x - 5)(x + 7).

So, the fraction can be rewritten as:
((x - 5)(x + 2)) / ((x - 5)(x + 7))

Just like with the numerical fraction, we can now see a common factor—the entire bracket (x - 5). By cancelling this common factor from the top and bottom, we are left with the simplified answer:

(x + 2) / (x + 7)

This process forms the core of today's lesson. Now, let's formalise the steps.

Simplifying Algebraic Fractions - GCSE Maths - Steps & Examples

The article "Simplifying Algebraic Fractions" from Third Space Learning clearly outlines the three-step process we just used.

Please read the section under the heading "How to simplify algebraic fractions". These three steps—calculate (if needed), factorise, and cancel—are the fundamental procedure you will always follow.

Part 3: Putting It All Together with Examples

Now that you've seen the method, let's apply it to a few examples that combine all the factorising skills you've developed. The following examples from the Third Space Learning article are excellent practice.

  • Linear ÷ Quadratic: (2x + 6) / (x^2 + 11x + 24)

    • Factorise: 2(x + 3) / (x + 3)(x + 8)
    • Cancel: The (x + 3) terms.
    • Result: 2 / (x + 8)

  • Quadratic ÷ Quadratic (Difference of Two Squares): (x^2 + 15x + 56) / (x^2 - 64)

    • This is a great example connecting directly to our last lesson.
    • Factorise: (x + 7)(x + 8) / (x + 8)(x - 8)
    • Cancel: The (x + 8) terms.
    • Result: (x + 7) / (x - 8)

As you can see, the simplification itself is straightforward. The real work, and where marks are won or lost, is in the factorising.

Part 4: An A*-level Exam Technique

For the IGCSE Extended paper, questions are often designed so that the numerator and denominator share a factor. You can use this to your advantage, especially with trickier quadratics. If you can factorise one part of the fraction easily, it gives you a massive clue for factorising the other part.

This "cheat method" is something that will help you solve problems much faster. The next video demonstrates this technique, which is perfect for your "quick solving" approach.

Hardest All of iGCSE Algebraic Fractions: What You Need To Know

This video from Ginger Mathematician shows how to use factorisation clues to simplify complex fractions quickly, a technique very useful for aiming for a top grade.

Watch the section where the presenter simplifies (2x^2 - 5x - 12) / (3x^2 - 12x). Notice how after factorising the denominator to 3x(x - 4), he states he's expecting to see an (x - 4) factor in the numerator. See how this makes factorising the more difficult 2x^2 - 5x - 12 much easier from this point on.

Part 5: A Critical Warning - Factors vs. Terms

One of the most common errors in algebra is incorrect cancellation. Remember that you can only cancel common factors, which are expressions that are multiplied. You can never cancel common terms, which are parts of an addition or subtraction.

  • Correct: (5 * (x+2)) / (5 * (x-3)) simplifies to (x+2) / (x-3). The 5 is a factor.
  • Incorrect: (5 + x) / (5 - y) CANNOT be simplified by cancelling the 5s. Here, 5 is a term.

A subtle version of this involves negative signs. Consider Practice Question 6 from the Third Space Learning article: (x^2 + 6x - 7) / (49 - x^2).

  • The top factorises to (x+7)(x-1).
  • The bottom is a difference of two squares, (7-x)(7+x).
    The fraction is (x+7)(x-1) / (7-x)(7+x).

Notice that (x+7) and (7+x) are identical, so they cancel. This leaves (x-1) / (7-x). These are not the same and cannot be cancelled directly. This is the final simplified form.

Conclusion

Today you've combined your factorising skills to tackle a new type of problem. Simplifying algebraic fractions is a process that turns messy expressions into manageable ones, and it's a skill you'll rely on heavily as we move into solving equations.

Key Takeaways:

  • To simplify an algebraic fraction, you must first factorise the numerator and the denominator completely.
  • Once factorised, you can cancel out any common factors that appear in both the numerator and the denominator.
  • Be vigilant about the difference between factors (multiplied) and terms (added/subtracted). Only factors can be cancelled.
  • Use the structure of the fraction as a clue. The factors of one part often reveal the factors of the other.

In our next lesson, we will move on to solving quadratic equations by factorising. You'll see how the expressions we've been manipulating today can be set to equal zero, allowing us to find the specific values of x that solve the equation.