📘Intermediate Algebra Unit 7 – Rational Expressions and Functions

Key Concepts

• Rational expressions are fractions with polynomials in the numerator and denominator
• Rational expressions are undefined when the denominator equals zero
• Simplifying rational expressions involves factoring the numerator and denominator and canceling common factors
• Operations with rational expressions include addition, subtraction, multiplication, and division
• When adding or subtracting rational expressions, a common denominator is required
• Multiplying rational expressions involves multiplying the numerators and denominators separately, then simplifying the result
• Dividing rational expressions is equivalent to multiplying by the reciprocal of the divisor
• Solving rational equations may involve cross-multiplication, factoring, or finding a common denominator

Simplifying Rational Expressions

• Factor the numerator and denominator completely
  • Identify the greatest common factor (GCF) of the terms in the numerator and denominator
  • Factor out the GCF from both the numerator and denominator
• Cancel common factors in the numerator and denominator
  • Identify any factors that appear in both the numerator and denominator
  • Divide out the common factors, canceling them from the numerator and denominator
• Ensure that the resulting expression has no common factors remaining in the numerator and denominator
• Avoid dividing by zero by identifying any values that would make the denominator equal to zero
• Simplify any remaining terms in the numerator and denominator
• Express the simplified rational expression in lowest terms

Operations with Rational Expressions

• Addition and subtraction require a common denominator
  • Find the least common multiple (LCM) of the denominators
  • Multiply each rational expression by the appropriate factor to obtain the common denominator
  • Add or subtract the numerators, keeping the common denominator
• Multiplication involves multiplying the numerators and denominators separately, then simplifying the result
  • Multiply the numerators together
  • Multiply the denominators together
  • Simplify the resulting rational expression by canceling common factors
• Division is performed by multiplying the first rational expression by the reciprocal of the second
  • Take the reciprocal of the divisor (the second rational expression)
  • Multiply the first rational expression by the reciprocal
  • Simplify the resulting rational expression
• Simplify the final result by factoring and canceling common factors

Solving Rational Equations

• Clear the denominators by multiplying both sides of the equation by the LCD
  • Find the least common denominator (LCD) of all the rational expressions in the equation
  • Multiply both sides of the equation by the LCD to eliminate the denominators
• Simplify the resulting equation by combining like terms and distributing
• Solve the simplified equation using appropriate methods (factoring, quadratic formula, etc.)
• Check the solutions by substituting them back into the original equation
  • Verify that the solutions do not make any denominators equal to zero
  • Confirm that the solutions satisfy the original equation
• Identify any extraneous solutions that may have been introduced during the solving process

Graphing Rational Functions

• Identify the domain of the rational function
  • Find the values of x that make the denominator equal to zero
  • Exclude these values from the domain
• Determine the vertical and horizontal asymptotes
  • Vertical asymptotes occur at x-values where the denominator equals zero
  • Horizontal asymptotes can be found by comparing the degrees of the numerator and denominator
    • If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0
    • If the degrees are equal, the horizontal asymptote is y = the leading coefficient of the numerator divided by the leading coefficient of the denominator
    • If the degree of the numerator is greater than the degree of the denominator by one, there is no horizontal asymptote; the function has an oblique asymptote
• Find the x- and y-intercepts
  • Set the numerator equal to zero and solve for x to find the x-intercepts
  • Set x equal to zero and solve for y to find the y-intercept
• Determine the behavior of the function near the asymptotes and intercepts
• Plot additional points as needed to sketch the graph

Applications of Rational Functions

• Rational functions can model various real-world situations, such as:
  • Rate problems (work rate, flow rate, etc.)
  • Mixture problems (concentrations of solutions)
  • Distance, speed, and time problems
• Identify the relevant variables and their relationships in the problem
• Set up a rational equation or function based on the given information
• Solve the equation or analyze the function to answer the question posed in the problem
• Interpret the results in the context of the original situation
• Verify that the solution makes sense and is reasonable given the context

Common Mistakes and Tips

• Remember that rational expressions are undefined when the denominator equals zero
• Always factor the numerator and denominator completely before canceling common factors
• When adding or subtracting rational expressions, find the LCD first and multiply each expression by the appropriate factor
• Be careful not to divide by zero when simplifying or solving rational expressions
• When solving rational equations, check for extraneous solutions by substituting the solutions back into the original equation
• Pay attention to the domain when graphing rational functions, and identify any asymptotes and intercepts
• In application problems, make sure to interpret the results in the context of the original situation

Practice Problems and Solutions

1. Simplify the rational expression: $\frac{6x^2 + 3x}{2x^2 + 5x - 3}$ Solution:

   • Factor the numerator: $3x(2x + 1)$
   • Factor the denominator: $(2x - 1)(x + 3)$
   • The simplified expression is $\frac{3x}{2x - 1}$

2. Perform the indicated operation and simplify: $\frac{2}{x + 1} - \frac{3}{x - 2}$ Solution:

   • Find the LCD: $(x + 1)(x - 2)$
   • Multiply each expression by the appropriate factor: $\frac{2}{x + 1} \cdot \frac{x - 2}{x - 2} - \frac{3}{x - 2} \cdot \frac{x + 1}{x + 1}$
   • Simplify: $\frac{2x - 4}{(x + 1)(x - 2)} - \frac{3x + 3}{(x + 1)(x - 2)}$
   • Combine the numerators: $\frac{2x - 4 - 3x - 3}{(x + 1)(x - 2)}$
   • Simplify: $\frac{-x - 7}{(x + 1)(x - 2)}$

3. Solve the rational equation: $\frac{2}{x - 1} + \frac{3}{x + 2} = \frac{5}{x^2 + x - 2}$ Solution:

   • Find the LCD: $(x - 1)(x + 2)$
   • Multiply both sides of the equation by the LCD: $(x - 1)(x + 2) \cdot \frac{2}{x - 1} + (x - 1)(x + 2) \cdot \frac{3}{x + 2} = (x - 1)(x + 2) \cdot \frac{5}{x^2 + x - 2}$
   • Simplify: $2(x + 2) + 3(x - 1) = 5$
   • Distribute: $2x + 4 + 3x - 3 = 5$
   • Combine like terms: $5x + 1 = 5$
   • Subtract 1 from both sides: $5x = 4$
   • Divide both sides by 5: $x = \frac{4}{5}$
   • Check the solution by substituting it back into the original equation

4. Graph the rational function: $f(x) = \frac{x + 1}{x - 3}$ Solution:

   • Identify the domain: x ≠ 3
   • Find the vertical asymptote: x = 3
   • Find the horizontal asymptote: y = 1 (degrees of numerator and denominator are equal)
   • Find the x-intercept: x = -1
   • Find the y-intercept: y = -1/3
   • Plot the asymptotes, intercepts, and additional points to sketch the graph