Simplifying Algebraic Fractions: Tips and Techniques

Introduction to Algebraic Fractions

Algebraic fractions are fractions where the numerator, denominator, or both contain algebraic expressions. This guide will help you understand how to work with these fractions effectively.

Simplifying Algebraic Fractions

To simplify an algebraic fraction:

  1. Factor the numerator and denominator
  2. Cancel out common factors

Example:

x2−4x−2=(x+2)(x−2)x−2=x+2\frac{x^2 - 4}{x - 2} = \frac{(x + 2)(x - 2)}{x - 2} = x + 2

Operations with Algebraic Fractions

Addition and Subtraction

  1. Find a common denominator
  2. Add or subtract the numerators
  3. Simplify the result

Example:

1x+1y=yxy+xxy=x+yxy\frac{1}{x} + \frac{1}{y} = \frac{y}{xy} + \frac{x}{xy} = \frac{x + y}{xy}

Multiplication

  1. Multiply the numerators
  2. Multiply the denominators
  3. Simplify the result

Example:

ab⋅cd=acbd\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}

Division

  1. Multiply by the reciprocal
  2. Simplify the result

Example:

ab÷cd=ab⋅dc=adbc\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c} = \frac{ad}{bc}

Complex Fractions

Complex fractions are fractions that contain fractions in the numerator or denominator. To simplify:

  1. Simplify the numerator and denominator separately

Example: