Algebraic Fractions

Fractions with numerators, denominators, or both as algebraic expressions, involving variables and constants.

Resources

Understanding Algebraic Fractions

What are Algebraic Fractions?

Algebraic fractions are similar to numerical fractions but include algebraic expressions in the numerator, denominator, or both. They are often used in algebra to simplify expressions and solve equations.

Basic Structure

An algebraic fraction has the form:

Ax + B

Cx + D

Where A, B, C, and D are constants and x is a variable.

Operations with Algebraic Fractions

1. Simplifying

To simplify an algebraic fraction, factor both the numerator and the denominator and then cancel out any common factors.

Example:

x² - 1

x + 1

Factor the numerator: (x - 1)(x + 1)

So, the fraction simplifies to: x - 1

2. Adding and Subtracting

To add or subtract algebraic fractions, find a common denominator and combine the numerators accordingly.

Example:

x / (x + 2)

+ 3 / (x + 2)

Since the denominators are the same, add the numerators:

Result: (x + 3) / (x + 2)

3. Multiplying and Dividing

To multiply algebraic fractions, multiply the numerators together and the denominators together. To divide, multiply by the reciprocal of the second fraction.

Example:

2 / (x + 1)

× 3 / (x - 1)

Multiply the numerators and denominators:

Result: (2 × 3) / ((x + 1) × (x - 1)) which simplifies to 6 / (x² - 1)

Key Points to Remember

Always factor expressions where possible to simplify fractions.
When adding or subtracting, ensure the denominators are the same.
When multiplying or dividing, handle each part of the fraction separately.