Math Procedures 1 - Algebraic Fractions

Algebraic Fractions Overview

• Algebraic fractions consist of a numerator and denominator, which can be polynomials.
• Key operations include addition, subtraction, multiplication, division, and finding least common denominators (LCD).

Factoring Expressions

• To simplify fractions, factor both the numerator and denominator.
• Example: Simplifying ( \frac{3x^2 - 14x + 8}{x^2 - 16} ) results in ( \frac{(3x - 2)(x - 4)}{(x - 4)(x + 4)} ), which simplifies to ( \frac{3x - 2}{x + 4} ) after canceling ( (x - 4) ).

Multiplying Algebraic Fractions

• Multiply fractions by multiplying the numerators together and the denominators together.
• This can also involve canceling common factors before performing the multiplication.

Cross Cancellation

• When multiplying multiple fractions, like terms can be canceled:
  • Example: In ( \frac{4x^2}{5y} \times \frac{10x}{y^2} \times \frac{y}{8x^2} ), canceling results in ( \frac{x}{y^2} ).

Dividing Algebraic Expressions

• Division of fractions involves inverting the second fraction and then multiplying, which facilitates simplification.

Adding/Subtracting Expressions

• Finding the LCD is crucial for adding or subtracting algebraic fractions.
• The LCD is determined by factoring each denominator and taking the highest power of each factor.

Steps to Find the LCD

• Factor all denominators thoroughly.
• The LCD is the product of distinct factors taken at their maximum occurrence found in any denominator.

Example LCD Calculation

• For ( \frac{1}{x^2 - x} ), ( \frac{1}{x^2 - 1} ), and ( \frac{1}{x^2} ):
  • Factored forms yield ( x(x - 1) ), ( (x + 1)(x - 1) ), and ( x^2 ).
  • The final LCD is ( x^2(x - 1)(x + 1) ).

Process for Adding/Subtracting Algebraic Fractions

• Step 1: Identify the LCD.
• Step 2: Rewrite fractions over the LCD.
• Step 3: Combine numerators over the LCD.
• Step 4: Simplify the resulting expression if possible.

Example of Simplification

• For ( \frac{y-3}{y-5} + \frac{y - 23}{y^2 - y - 20} ), factor and combine:
  • LCD found as ( (y-5)(y+4) ).
  • Final expression after simplification leads to ( \frac{y + 7}{y + 4} ) with the condition that ( y \neq 5 ).

General Tips

• Ensure thorough factorization before performing operations to identify possible cancellations.
• Always check for undefined values (e.g., denominators equating to zero) before simplifying fractions.