1.0+P.6+Rational+Expressions

Rational Expressions

Overview of Topics

  • Specify numbers that must be excluded from the domain of a rational expression.

  • Simplify rational expressions.

  • Multiply rational expressions.

  • Divide rational expressions.

  • Add and subtract rational expressions.

  • Simplify complex rational expressions.

Definitions

  • Rational Expression: A fraction where both the numerator and denominator are polynomials.

  • Domain: All real numbers for which the expression is defined, excluding values where the denominator is zero (i.e., ( d
    eq 0 )).

  • Excluded Values: Specific values of the variable that make the denominator zero and thus are not included in the domain of the rational expression.

Excluded Values from Domains

Example Problem

Exercise 1: Find all numbers that must be excluded from the domain of a rational expression:Given expression: ( \frac{7}{2 - 5 - 14} )

  • Step 1: Identify values that make the denominator zero:( 2 - 5 - 14 = 0 )

Calculation

  • Step 2: Solve for x:( 2 - 5 - 14 = -17 ) (no excluded values in this case)

Simplifying Rational Expressions

  • A rational expression is in simplest form when the numerator and denominator have no common factors other than 1 or -1.

Steps to Simplify

  1. Factor the numerator and denominator completely.

  2. Identify all excluded values from the original expression.

  3. Cancel any common factors found in the numerator and denominator.

  4. Note that excluded values from the original expression remain excluded in the simplified expression.

Simplification Exercises

Exercise 2: Simplify: ( \frac{3 - 32}{2 + 5 - 24} )

Multiplying and Dividing Rational Expressions

Multiplication Steps

  1. Factor all numerators and denominators completely.

  2. Simplify expressions beforehand if possible.

  3. Multiply the numerators and the denominators, keeping them in factored form to simplify later if needed.

Division Steps

  1. Invert the second rational expression (take the reciprocal).

  2. Multiply the resulting expressions.

Division and Multiplication Exercises

Exercise 3: Simplify: ( \frac{x + 3}{2 - 6} \times \frac{2 - 4}{2 + 6 + 9} ) Exercise 4: Simplify: ( \frac{2}{2 - 2} + 1 = \frac{22 + 3}{3} )

Adding and Subtracting Rational Expressions

With Same Denominator

Steps:

  1. Add or subtract the numerators.

  2. Place over the common denominator.

  3. Simplify the fraction if applicable.

  4. Check for any excluded values during simplification.

Exercise 5

Exercise 5: Simplify: ( \frac{3 + 2}{+1} + 1 )

Finding the Least Common Denominator (LCD)

Steps

  1. Determine the least common denominator when adding or subtracting rational expressions.

  2. Factor each denominator completely.

  3. Use the greatest number of times each unique factor appears within the denominators.

  4. Multiply the listed factors to find the LCD.

Example Exercise

Exercise 6: Find LCD for: ( 2 - 6 + 9 ) and ( 2 - 9 )

Adding/Subtracting Rational Expressions

Process

  1. Identify the LCD.

  2. Write equivalent expressions with the LCD as the common denominator.

  3. Add or subtract the numerators and place the total over the LCD.

  4. Simplify the expression if possible to its lowest terms.

Exercise 7

Exercise 7: Simplify: ( \frac{-4}{2 - 10 + 25} \cdot (2 - 10) )

Complex Rational Expressions

  • A complex rational expression contains rational expressions in either the numerator or the denominator. Must be simplified such that neither contains any rational expressions.

  • Example: ( 1 + \frac{h - 1}{h} )

Simplifying Complex Rational Expressions

Steps

  1. Find the Least Common Multiple (LCM) of all denominators in both the numerator and the denominator.

  2. Multiply both the numerator and denominator by the LCM.

  3. Simplify to lowest terms after cancellation of common factors and rational expressions.

Exercises

Exercise 8: Simplify: ( \frac{1}{7} )Exercise 9: Simplify: ( \frac{3}{6} \times (6283) )