8.1 Simplify Rational Expressions

Rational expressions are like fractions, but with algebraic terms instead of numbers. They're a key part of algebra, helping us solve complex problems and model real-world situations.

Simplifying these expressions is crucial for easier calculations and clearer understanding. We'll learn how to factor, cancel out common terms, and handle tricky situations like undefined values and complex fractions.

Simplifying Rational Expressions

Introduction to Rational Expressions

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• A rational expression is an algebraic fraction where both the numerator and denominator are polynomials
• Simplification of rational expressions involves reducing them to their simplest form
• Understanding variable restrictions is crucial when working with rational expressions

Undefined values in rational expressions

• Rational expressions become undefined when the denominator equals zero
  • Set the denominator equal to zero and solve for the variable to find the undefined values
• $\frac{2x+1}{x-5}$ is undefined when $x-5=0$, so $x=5$ is the undefined value

Evaluation of rational expressions

• Substitute the given value for the variable and simplify to evaluate a rational expression
  • Expression is undefined if the substituted value results in the denominator being zero
• Factor the numerator and denominator completely to simplify a rational expression
  • Cancel out common factors in the numerator and denominator
  • Simplified expression is formed by the remaining factors in the numerator and denominator
• Simplify $\frac{x^2-9}{x+3}$
  1. Factor the numerator: $\frac{(x+3)(x-3)}{x+3}$
  2. Cancel the common factor $(x+3)$: $x-3$

Simplification with opposite factors

• Opposite factors are binomials with the same terms but opposite signs $(x+2)$ and $(x-2)$
• Rational expressions containing opposite factors can be canceled out
  • Simplified expression will have a negative sign in the numerator or denominator
• Simplify $\frac{x-4}{x+4}$
  • Numerator and denominator are opposite factors
  • Cancel them out, resulting in $-1$

Techniques for complex rational expressions

• Complex rational expressions contain rational expressions in the numerator, denominator, or both
• Simplify complex rational expressions by:
  1. Factoring the numerator and denominator completely
  2. Identifying and canceling out common factors
  3. Combining the remaining factors to form the simplified expression
• Simplify $\frac{\frac{x-3}{x+3}}{\frac{x-1}{x+1}}$
  1. Multiply the numerator and denominator by the LCD of $(x+3)(x+1)$
     • $\frac{(x-3)(x+1)}{(x-1)(x+3)}$
  2. Factor the numerator and denominator: $\frac{(x+1)(x-3)}{(x+3)(x-1)}$
  3. Cancel out common factors: $\frac{x-3}{x-1}$