Algebraic Fractions (Edexcel GCSE Maths): Revision Notes

Algebraic fractions

Working with algebraic fractions follows the same principles as working with numerical fractions, but now we're dealing with expressions that contain letters (variables) as well as numbers. The good news is that all the rules you've learned for numerical fractions apply here too - you just need to be more careful with your algebra!

Simplifying algebraic fractions

When simplifying algebraic fractions, your goal is to cancel out common factors that appear in both the numerator and denominator. This process makes the fraction easier to work with and often reveals its simplest form.

The key steps for simplifying are:

• Look for common factors in the numerator and denominator
• Cancel these common factors
• If needed, factorise expressions first to reveal hidden common factors

Let's look at how this works in practice. For simple expressions like those involving coefficients and variables, you can cancel numerical coefficients and subtract the powers of like variables. For example, when you have the same variable in both the numerator and denominator, you can reduce the powers by subtracting the smaller power from the larger one.

For more complex expressions, you might need to factorise first. This is particularly useful when you have quadratic expressions or differences of squares. The difference of two squares pattern (often remembered as "DOTS") is especially helpful - it allows you to factor expressions like $x^2 - 16$ into $(x + 4)(x - 4)$.

Once you've factorised both the numerator and denominator, look for identical factors that appear in both parts. These can be cancelled out, leaving you with a much simpler expression.

Multiplying algebraic fractions

Multiplying algebraic fractions is actually the most straightforward operation. The rule is simple and direct.

To multiply two algebraic fractions, multiply the numerators together and multiply the denominators together. However, there's a useful shortcut: you can cancel common factors before multiplying, which often makes the calculation much easier.

The mathematical rule can be expressed as: $\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$

Dividing algebraic fractions

Division of algebraic fractions follows the same "flip and multiply" rule that you use with numerical fractions.

When dividing by a fraction, you multiply by its reciprocal instead. This means you flip the second fraction (swap its numerator and denominator) and then multiply. After flipping, you can treat the problem as a multiplication and apply all the same techniques, including cancelling common factors before multiplying.

Remember that division by a fraction is the same as multiplication by its reciprocal, so $a \div \frac{b}{c} = a \times \frac{c}{b}$.

Adding and subtracting algebraic fractions

Adding and subtracting algebraic fractions is more challenging because you need to work with common denominators. Unlike multiplication and division, you can't simply combine numerators and denominators directly.

The process involves several steps:

• Find a common denominator for all fractions
• Rewrite each fraction with this common denominator
• Add or subtract the numerators
• Keep the common denominator

The common denominator is typically the product of all the different denominators, though sometimes you can use a simpler expression if the denominators share common factors.

When working with more complex denominators, you'll need to multiply each fraction by appropriate factors to achieve the common denominator. For instance, if you're adding fractions with denominators $(x + 3)$ and $(x - 2)$, your common denominator will be $(x + 3)(x - 2)$.

Each fraction must be multiplied by the factors it's missing from the common denominator. The first fraction gets multiplied by $\frac{(x-2)}{(x-2)}$, and the second by $\frac{(x+3)}{(x+3)}$. Once both fractions have the same denominator, you can add the numerators and simplify if possible.