Notes on Simplifying and Adding Rational Expressions

Objectives of Simplifying Rational Expressions

• Importance: Simplifying rational expressions is crucial for solving rational equations and presenting simplified fractions for easier computation.
• Examples of Simplification:
  • Instead of writing $\frac{1}{2}$ as $\frac{2}{4}$, we prefer the simplest form.
  • Considerations for probability or odds in gambling: sometimes not desired to simplify.

Steps to Simplify Rational Expressions

1. Factor the Numerator and Denominator:
   • Identify and factor out common terms in the numerator and denominator.
2. Cancel Common Factors:
   • If common factors exist in both numerator and denominator, they can be canceled out.
3. Write the Entire Expression:
   • Combine the simplified fractions and ensure the expression is fully reduced.

Example Problems

Example 1: Simplifying a Complex Expression

• Given Expression: $\frac{5}{2} \cdot \frac{x - 1}{x + 1}$ + $\frac{1}{x + 1}$
• Steps:
  1. Numerator:
  • Simplify: $5 \cdot (x + 1) \cdot (x - 1)$
  1. Denominator:
  • Simplify: $x(x + 1) + (x - 1)$ results in $x^2 + x - 1$
  1. Reformulate as a Division:
  • Final Expression: $\frac{3x + 7}{(x - 1)(x + 1)}$

Example 2: Simplifying a Fraction

• Problem: Simplify $\frac{30t^3}{50t^5}$
• Steps:
  1. Factor:
  • Numerator: $30 = 2 \cdot 3 \cdot 5$
  • Denominator: $50 = 2 \cdot 5^2$
  1. Cancel Common Factors:
  • Cancel out: 2, 5, and t
  1. Simplify: Final expression: $\frac{3}{5t^2}$

Example 3: Factoring Quadratics

• Given Expression: $\frac{x^2 - 8x + 12}{x^2 - 5x + 6}$
• Steps:
  1. Numerator: Factor to $(x - 6)(x - 2)$
  2. Denominator: Factor to $(x - 3)(x - 2)$
  3. Cancel: Common factor is $(x - 2)$ results in $\frac{x - 6}{x - 3}$

Adding and Subtracting Rational Expressions

Procedure

1. Distribute Negative Signs: When subtracting, distribute negative signs through the terms.
2. Find Common Denominator: Necessary to add or subtract fractions.
3. Combine Like Terms: Ensure you group like terms appropriately in the numerator.
4. Simplify: Write the final sum or difference and simplify if possible.

Example of Addition/Subtraction

• Add and Simplify: $\frac{10x + 8}{4} + \frac{52 + 4}{4}$
  1. Simplify the Numerator: Combine the like terms from both numerators.
  2. Combine Results: Rewrite in simplest form.

Final Remarks

• Always seek the simplest form, factoring where possible and canceling common terms where applicable to ensure clarity in responses and accuracy in answers to rational expressions.