Algebraic Fractions

Study Notes

Algebraic fractions involve expressions where the numerator and/or the denominator are algebraic expressions. You will learn to simplify, add, subtract, multiply, and divide these fractions.

• Algebraic Fraction — a fraction where the numerator and/or the denominator is an algebraic expression. Example: $\frac{x+2}{x-3}$
• Lowest Common Multiple (LCM) — the smallest multiple that is exactly divisible by each number in a set. Example: The LCM of 7 and 2 is 14.
• Factorising — rewriting an expression as a product of its factors. Example: $x^2 - 9 = (x+3)(x-3)$
• Common Denominator — a shared multiple of the denominators of two or more fractions. Example: For $\frac{1}{x}$ and $\frac{1}{2y}$, a common denominator is $2xy$.

Exam Tips

Key Definitions to Remember

• Algebraic Fraction: A fraction with algebraic expressions in the numerator or denominator.
• Lowest Common Multiple (LCM): The smallest number that is a multiple of two or more numbers.
• Factorising: Breaking down an expression into a product of simpler expressions.

Common Confusions

• Forgetting to factorise before simplifying.
• Trying to simplify through addition or subtraction terms.

Typical Exam Questions

• How do you add $\frac{1}{x}$ and $\frac{1}{2y}$? Find a common denominator, then add.
• How do you simplify $\frac{x^2 - 9}{x^2 - 3x}$? Factorise both the numerator and denominator, then cancel common terms.
• What is the LCM of 3x and 6y? The LCM is 6xy.

What Examiners Usually Test

• Ability to find and use the lowest common multiple for addition and subtraction.
• Simplifying complex algebraic fractions by factorising and cancelling like terms.
• Understanding when to use common denominators.