Algebraic Fractions

Study Notes

Algebraic fractions involve fractions where the numerator and/or the denominator contain algebraic expressions.

• Lowest Common Multiple (LCM) — the smallest multiple that is exactly divisible by each denominator. Example: The LCM of x and 2y is 2xy.
• Addition/Subtraction of Algebraic Fractions — process of making denominators the same before adding or subtracting numerators. Example: To add $\frac{1}{x}$ and $\frac{1}{2y}$, find the LCM of x and 2y, which is 2xy.
• Multiplication/Division of Algebraic Fractions — multiply or divide directly without making denominators the same. Example: $\frac{3}{4} \times \frac{2}{3} = \frac{6}{12}$.
• Simplifying Algebraic Fractions — factorize numerators and denominators, then cancel common factors. Example: Simplify $\frac{x^2 - 1}{x + 1}$ by factoring to $\frac{(x - 1)(x + 1)}{x + 1}$ and cancelling $x + 1$.

Exam Tips

Key Definitions to Remember

• Lowest Common Multiple (LCM) is the smallest multiple shared by denominators.
• Simplifying involves factorizing and cancelling common factors.

Common Confusions

• Forgetting to find the LCM for addition or subtraction.
• Trying to make denominators the same for multiplication or division.

Typical Exam Questions

• How do you add $\frac{1}{x}$ and $\frac{1}{2y}$? Find the LCM of x and 2y, which is 2xy, then add.
• How do you simplify $\frac{x^2 - 1}{x + 1}$? Factorize to $\frac{(x - 1)(x + 1)}{x + 1}$ and cancel $x + 1$.
• What is the result of $\frac{3}{4} \times \frac{2}{3}$? Multiply directly to get $\frac{6}{12}$.

What Examiners Usually Test

• Ability to find and use the LCM for addition and subtraction.
• Correct simplification of algebraic fractions.
• Understanding when to factorize and cancel terms.