Rational Expressions and Equations

A rational expression is a ratio of two polynomials. Working with rational expressions is very similar to working with standard numeric fractions, but variables are involved. They appear frequently when dealing with rates, work problems, and inverse variation.

Domain Restrictions

Because division by zero is undefined, any value of the variable that makes the denominator equal to zero must be excluded from the domain.

Finding the Domain

  • Set the polynomial in the denominator equal to zero.
  • Solve for the variable.
  • Exclude those values from the domain.

Interactive Asymptote Visualizer

Explore how vertical and horizontal asymptotes behave in a rational function when altering the numerator and denominator coefficients.

Operations on Rational Expressions

Multiplication and Division

  • Multiplication: Factor all numerators and denominators completely, then cancel common factors before multiplying across.
  • Division: Multiply the first expression by the reciprocal of the second expression (flip the second fraction), then proceed as multiplication.

Addition and Subtraction

  • Same Denominators: Add or subtract the numerators and keep the common denominator.
  • Different Denominators: Find the Least Common Denominator (LCD), convert each fraction to an equivalent fraction with the LCD, then add/subtract numerators.

Complex Fractions

A complex fraction is a rational expression that contains fractions within its numerator, denominator, or both.

Simplification Strategies

  • Method 1 (Single Fraction): Simplify the numerator into a single fraction and the denominator into a single fraction. Then multiply the top fraction by the reciprocal of the bottom fraction.
  • Method 2 (LCD Method): Find the Least Common Denominator (LCD) of all minor fractions within the expression. Multiply the main numerator and main denominator by this LCD to instantly clear all inner fractions.

Solving Rational Equations

To solve equations involving rational expressions, the most efficient method is often to clear the denominators by multiplying every term by the LCD.

Extraneous Solutions

Always check your final answers against the original domain restrictions. A solution is extraneous if it makes any denominator in the original equation equal to zero.

Partial Fraction Decomposition

Partial fraction decomposition is the reverse of adding rational expressions. It breaks a complex rational expression into a sum of simpler fractions. This technique is incredibly useful in integral calculus, Laplace transforms, and control systems engineering.

Decomposition Rules

For a proper rational expression P(x)Q(x)\frac{P(x)}{Q(x)} (where degree of P(x)P(x) <\lt degree of Q(x)Q(x)), factor Q(x)Q(x) completely, then set up the partial fractions based on the factors:
  • Distinct Linear Factors: For each factor (ax+b)(ax + b), add a term Aax+b\frac{A}{ax + b}.
  • Repeated Linear Factors: For (ax+b)n(ax + b)^n, add A1ax+b+A2(ax+b)2+⋯+An(ax+b)n\frac{A_1}{ax + b} + \frac{A_2}{(ax + b)^2} + \dots + \frac{A_n}{(ax + b)^n}.
  • Irreducible Quadratic Factors: For (ax2+bx+c)(ax^2 + bx + c), add a term Ax+Bax2+bx+c\frac{Ax + B}{ax^2 + bx + c}.

Rational Inequalities

Rational inequalities take the form P(x)Q(x)>0\frac{P(x)}{Q(x)} > 0, ≥0\ge 0, <0< 0, or ≤0\le 0. Similar to polynomial inequalities, we use a sign chart to analyze intervals.

Solving Strategy

  • Step 1: Bring all terms to one side of the inequality. The other side must be exactly zero.
    • DO NOT multiply both sides by an expression involving a variable because you do not know if the expression is positive or negative.
  • Step 2: Combine fractions so the non-zero side is a single rational expression.
  • Step 3: Factor the numerator and the denominator completely. The roots of the numerator are the x-intercepts, and the roots of the denominator are the vertical asymptotes. Both sets of roots are the critical points.
  • Step 4: Plot all critical points on a number line. This creates test intervals.
  • Step 5: Determine the sign of the rational expression within each interval using test values. Use a sign chart to evaluate where the expression satisfies the original inequality.

Domain Restrictions in Inequalities

Always use open intervals (parentheses) for any critical point derived from the denominator, because the function is undefined there. Only critical points from the numerator can be included with a closed bracket [] if the inequality is non-strict (≥\ge or ≤\le).
Key Takeaways
  • Factor First: Always factor numerators and denominators completely before doing anything else.
  • Domain Restrictions: Denominators can never equal zero. Note restrictions before cancelling factors.
  • Addition/Subtraction: You MUST have a common denominator.
  • Multiplication/Division: You do NOT need a common denominator.
  • Extraneous Roots: Multiplying by variables can introduce false solutions. Always check your answers.
  • Rational Inequalities: Both the roots of the numerator and the roots of the denominator act as critical points on your sign chart. Never include values that make the denominator zero in your final answer.
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