Algebraic Fractions

Definition and Introduction

Algebraic fractions are simply fractions where the numerator and/or the denominator are algebraic expressions.
Like numerical fractions, they can be added, subtracted, multiplied, and divided.

Simplifying Algebraic Fractions

Simplification is the process of reducing an algebraic fraction to its simplest form.
This usually involves factorising the numerator and the denominator and cancelling any common factors.
This process also applies when simplifying complex algebraic fractions where the numerator and the denominator could be expressions.

Adding and Subtracting Algebraic Fractions

When adding or subtracting algebraic fractions, as with numerical fractions, the denominators must be the same.
Make the denominators the same by finding the least common multiple (LCM) and adjusting the numerators accordingly.
Once the denominators are the same, the fractions can be added or subtracted by combining the numerators.

Multiplying and Dividing Algebraic Fractions

Multiplying algebraic fractions involves multiplying the numerators together and the denominators together.
Dividing algebraic fractions requires flipping (finding the reciprocal of) the second fraction and then multiplying.
Aim to simplify or cancel down before multiplying or dividing to make calculations easier.

Solving Equations with Algebraic Fractions

Equations with algebraic fractions can be solved by clearing the fractions and then solving as a linear or quadratic equation.
To clear the fractions, multiply every term by the LCM of the denominators.
Remember to check your solutions in the original equation as there may be excluded values for which the original fraction is undefined.

Key Points

Algebraic fractions are fractions where the numerator and/or the denominator are algebraic expressions.
Simplify fractions by factorising and cancelling common factors.
Add or subtract fractions by ensuring the denominators are the same and then combining the numerators.
Multiply fractions by multiplying the numerators together and the denominators together, divide by turning the second fraction upside down and multiplying.
Solve equations with algebraic fractions by multiplying every term by the LCM of the denominators to clear the fractions and then solve as a normal equation.

Algebraic fractions although challenging, follow the same principles as their numerical counterparts. Practice will make mastering this concept much easier.