Algebraic Fraction

Grade 12 Math - Algebraic Fractions

Lesson Objectives

Understand how to simplify algebraic fractions.
Perform operations (addition, subtraction, multiplication, and division) on algebraic fractions.
Apply factorization techniques in simplification.

Lesson Introduction

Algebraic fractions are fractions that contain algebraic expressions in the numerator, denominator, or both. Simplifying algebraic fractions often involves:

Factoring expressions
Cancelling common factors
Finding the least common denominator (LCD) for addition/subtraction

Core Lesson Content

Worked Examples

Example 1: Simplify:

\frac{x^2 - 9}{x^2 - x - 6}

Factor:

\frac{(x - 3)(x + 3)}{(x - 3)(x + 2)} = \frac{x + 3}{x + 2}

Example 2: Simplify:

\frac{2x^2 + 5x + 2}{x^2 - 1}

Factor:

\frac{(2x + 1)(x + 2)}{(x - 1)(x + 1)}

Example 3: Multiply:

\frac{x^2 - 4}{x^2 - x - 6} \cdot \frac{x + 2}{x - 2}

Factor and cancel:

\frac{(x - 2)(x + 2)}{(x - 3)(x + 2)} \cdot \frac{x + 2}{x - 2} = \frac{1}{x - 3}

Example 4: Divide:

\frac{x^2 - 1}{x^2 + 2x + 1} \div \frac{x - 1}{x + 1}

Rewrite as multiplication:

\frac{(x - 1)(x + 1)}{(x + 1)^2} \cdot \frac{x + 1}{x - 1} = \frac{1}{x + 1}

Example 5: Add:

\frac{2}{x} + \frac{3}{x + 1}

LCD =

x(x + 1)

, so:

\frac{2(x + 1) + 3x}{x(x + 1)} = \frac{2x + 2 + 3x}{x(x + 1)} = \frac{5x + 2}{x(x + 1)}

Example 6: Subtract:

\frac{3x}{x^2 - 1} - \frac{2}{x - 1}

Factor denominator:

x^2 - 1 = (x - 1)(x + 1)

. LCD =

(x - 1)(x + 1)

Use equivalent fractions and simplify.

Example 7: Simplify:

\frac{x^2 + x - 6}{x^2 - 4x + 3}

Factor:

\frac{(x + 3)(x - 2)}{(x - 3)(x - 1)}

Example 8: Simplify:

\frac{4x^2 - 9}{2x + 3}

Use difference of squares:

\frac{(2x - 3)(2x + 3)}{2x + 3} = 2x - 3

Example 9: Add:

\frac{1}{x + 2} + \frac{2}{x - 2}

LCD =

(x + 2)(x - 2)

, so:

\frac{x - 2 + 2(x + 2)}{(x + 2)(x - 2)} = \frac{x - 2 + 2x + 4}{x^2 - 4} = \frac{3x + 2}{x^2 - 4}

Example 10: Divide:

\frac{x^2 - 2x}{x^2 - 1} \div \frac{x - 2}{x - 1}

Rewrite and simplify using multiplication.

Exercises

[WAEC] Simplify: $\frac{x^2 - 16}{x^2 - 4x}$ [Past Question]
Multiply: $\frac{2x + 4}{x^2 - 4} \cdot \frac{x - 2}{2x}$
[NECO] Divide: $\frac{x^2 - 9x + 20}{x^2 - 25} \div \frac{x - 4}{x - 5}$ [Past Question]
Add: $\frac{3}{x - 1} + \frac{2}{x + 1}$
Simplify: $\frac{x^2 - x - 6}{x^2 - 4}$
Subtract: $\frac{x + 2}{x^2 - x - 6} - \frac{1}{x - 3}$
[WASSCE] Simplify: $\frac{3x^2 - 27}{x^2 + 2x - 3}$ [Past Question]
Multiply: $\frac{x^2 - 9}{x^2 - x - 6} \cdot \frac{x + 2}{x - 3}$
Divide: $\frac{x^2 - 4}{x + 2} \div \frac{x - 2}{x - 1}$
[NECO] Add: $\frac{x + 1}{x^2 - 1} + \frac{x - 1}{x^2 - 1}$ [Past Question]

Conclusion/Recap

To succeed with algebraic fractions, always begin by factoring and simplifying where possible. For addition and subtraction, find the LCD. For multiplication and division, factor and cancel before simplifying. Mastery comes with practice!