Subtracting Rational Expressions

• Rational Expressions

  • When adding or subtracting rational expressions, the first step is to ensure denominators have a common denominator (LCD).

• Finding the LCD

  • Example:
  • Given expressions:
    1. \frac{2}{y(y+4)}
    2. \frac{4}{4(y+4)}
  • LCD derived: 4y(y+4)

• Subtracting Rational Expressions

  • If denominators are different, rewrite them to have the common denominator.
  • For example:
  • From the fractions \frac{x^2 - 2}{2x^2 - x - 3} and \frac{3 - 2x}{(x - 2)}, the LCD is \left(2x - 3\right)(x + 1).
  • Ensure to multiply one of the fractions by 1 in the form \frac{denominator}{denominator} to match the LCD.

• Steps for Subtracting

  1. Rewrite both fractions to have the same denominator.
  2. Combine the numerators over the common denominator.
  3. Simplify, if possible.

• Example Calculation

  • Compute:
  • \frac{x^2 - 2 + (2 - x)(x + 1)}{(2x - 3)(x + 1)}
  • Combine like terms in the numerator: \left(x^2 + x - 2 - x + 2\right) to get a simplified form.

Rational Expressions
When adding or subtracting rational expressions, the first step is to ensure denominators have a common denominator (LCD).

Finding the LCD

• Example:
  • Given expressions:
  1. \frac{2}{y(y+4)}
  2. \frac{4}{4(y+4)}
• LCD derived: 4y(y+4)

Subtracting Rational Expressions

• If denominators are different, rewrite them to have the common denominator.
• For example:
  • From the fractions \frac{x^2 - 2}{2x^2 - x - 3} and \frac{3 - 2x}{(x - 2)}, the LCD is \left(2x - 3\right)(x + 1).
• Ensure to multiply one of the fractions by 1 in the form \frac{denominator}{denominator} to match the LCD.

Steps for Subtracting

1. Rewrite both fractions to have the same denominator.
   • This may involve factoring polynomials in the denominators.
2. Combine the numerators over the common denominator.
3. Simplify the result, if possible, by factoring or reducing.

Example Calculation

• Compute:
  \frac{x^2 - 2 + (2 - x)(x + 1)}{(2x - 3)(x + 1)}
• Combine like terms in the numerator:
  \left(x^2 + x - 2 - x + 2\right) to get a simplified form, which can ultimately be expressed as \frac{x^2}{(2x - 3)(x + 1)} after cancellation of terms where applicable.

Important Note
Always check for restrictions in the original expressions before combining or simplifying, as certain values for the variable may lead to undefined expressions. Additionally, when working with rational expressions, ensure that expressions are simplified to their lowest terms