Lecture Notes: Simplifying Algebraic Fractions

Introduction to Simplifying Fractions

• Objective: Simplify algebraic expressions by finding a common denominator.
• Basic Concept: To add or subtract fractions, find a common denominator.
  • Example: For (\frac{1}{3} + \frac{1}{5}), find a common denominator.

Steps to Simplify

1. Factorize Denominators

   • Ensure all parts of the expression are factorized.
   • Example given: Factorize trinomials to find common terms.

2. Determine the Lowest Common Denominator (LCD)

   • Factorize expressions and identify common terms.
   • Combine all different parts to form the LCD.
   • Example given:
     • Expressions: (x + 7), (x^2 - x - 6)
     • Factorized to: ((x + 2)(x - 3))

3. Make Denominators the Same

   • Multiply expressions to have the same denominator as the LCD.
   • Ensure to apply changes to both numerator and denominator.
   • Use brackets to maintain expression integrity.

4. Simplifying the Expression

   • Combine all parts under one common denominator.
   • Simplify the numerator by expanding and combining like terms.

Example Problem Walkthrough

• Factorization Process: For (6), choose factor pairs and identify which pair fits the middle term of the trinomial.
• Finding the LCD: Combine distinct factors from each part.
  • Example: ((x + 2)(x - 3)) and other common factors are included.

Final Steps in Simplification

1. Expand Numerator

   • Multiply through brackets and simplify expression.
   • Combine like terms in the numerator.

2. Factor and Simplify Further

   • Check if the numerator can be factorized further.
   • Cancel out common terms between numerator and denominator.

3. Final Answer

   • Reduce to simplest form.
   • Example: (-\frac{1}{x - 3})

Key Points

• Always factorize denominators first.
• Use brackets to manage the scope of multiplication.
• Maintain a structured approach: Identify, Execute, Simplify.
• Pay attention to signs when expanding and simplifying.
• Verify if further simplification is possible through factorization and cancellation.