College Algebra - Week 9

Rational Expressions

Rational expressions are fractions with polynomials in the numerator and denominator. Every skill you learned for numeric fractions extends here: simplifying, multiplying, dividing, finding common denominators, and adding or subtracting. This week also introduces rational equations and the concept of extraneous solutions — values that appear to solve an equation but make the original expression undefined.

Learning Objectives

- Identify the domain of a rational expression by excluding values that make the denominator zero. - Simplify rational expressions by factoring and canceling common factors. - Multiply and divide rational expressions. - Add and subtract rational expressions with like and unlike denominators. - Simplify complex fractions. - Solve rational equations and check for extraneous solutions. - Solve work-rate and distance-rate problems using rational equations.

1. What Is a Rational Expression?

A rational expression is a ratio of two polynomials: P(x)/Q(x) where Q(x) is not zero. Examples: (x + 1)/(x - 3), (2x^2 - x)/(x^2 + 5x + 6). Domain: all real numbers EXCEPT values making the denominator equal to zero. Example: Find the domain of (x + 4)/(x^2 - x - 6). Factor denominator: (x - 3)(x + 2). Exclude x = 3 and x = -2. Domain: all real numbers except 3 and -2.

2. Simplifying Rational Expressions

Factor both numerator and denominator completely, then cancel common factors. WARNING: Cancel factors, NOT terms. (x + 3)/(x + 5) cannot be simplified. But (x + 3)^2/((x + 3)(x - 1)) = (x + 3)/(x - 1). Example 1: Simplify (x^2 - 9)/(x^2 + 5x + 6). Num: (x+3)(x-3). Den: (x+2)(x+3). Cancel (x+3): (x-3)/(x+2). Domain: x ≠ -3 and x ≠ -2. Example 2: Simplify (2x^2 - 2x)/(4x - 4). Num: 2x(x-1). Den: 4(x-1). Cancel (x-1): 2x/4 = x/2. Domain: x ≠ 1.

3. Multiplying Rational Expressions

Multiply numerators together and denominators together. Simplify before multiplying if possible. Example: Multiply (x^2 - 4)/(x + 3) * (x^2 + 5x + 6)/(x - 2). Factor first: (x+2)(x-2)/(x+3) * (x+3)(x+2)/(x-2) Cancel (x-2) and (x+3): = (x+2)(x+2) = (x+2)^2 Always factor before multiplying to avoid messy arithmetic.

4. Dividing Rational Expressions

Dividing by a fraction equals multiplying by its reciprocal. A/B ÷ C/D = A/B * D/C Example: Divide (x^2 - 25)/(x + 3) ÷ (x - 5)/(x^2 + 6x + 9). Flip and multiply: = (x^2-25)/(x+3) * (x^2+6x+9)/(x-5) Factor: = (x+5)(x-5)/(x+3) * (x+3)^2/(x-5) Cancel (x-5) and (x+3): = (x+5)(x+3)

5. Adding and Subtracting with Like Denominators

If denominators are the same, add or subtract numerators and keep the denominator. Example 1: 3/(x+2) + 7/(x+2) = 10/(x+2). Example 2: (2x + 1)/(x^2 - 4) - (x - 3)/(x^2 - 4) = (2x + 1 - (x - 3))/(x^2 - 4) = (x + 4)/(x^2 - 4) = (x + 4)/((x+2)(x-2)) Note: when subtracting, distribute the minus sign to every term in the second numerator.

6. Finding the LCD for Unlike Denominators

Step 1: Factor all denominators completely. Step 2: The LCD contains each unique factor raised to its highest power. Example: Find the LCD of 1/(x^2 - 1) and 1/(x^2 - x). Factor: x^2 - 1 = (x+1)(x-1) and x^2 - x = x(x-1). LCD = x(x+1)(x-1). Example 2: Find the LCD of 2/x and 3/(x+5). LCD = x(x+5).

7. Adding and Subtracting with Unlike Denominators

Example: Compute 3/x + 2/(x+5). LCD = x(x+5). 3/x = 3(x+5)/(x(x+5)). 2/(x+5) = 2x/(x(x+5)). Add: (3(x+5) + 2x)/(x(x+5)) = (3x+15+2x)/(x(x+5)) = (5x+15)/(x(x+5)). Factor numerator: 5(x+3)/(x(x+5)). No common factors to cancel. Final answer: 5(x+3)/(x(x+5)).

8. Subtracting Rational Expressions

Example: Compute x/(x-3) - 2/(x+1). LCD = (x-3)(x+1). First fraction: x(x+1)/((x-3)(x+1)). Second fraction: 2(x-3)/((x-3)(x+1)). Subtract: = (x(x+1) - 2(x-3))/((x-3)(x+1)) = (x^2 + x - 2x + 6)/((x-3)(x+1)) = (x^2 - x + 6)/((x-3)(x+1)) This numerator does not factor nicely, so this is the final answer.

9. Complex Fractions

A complex fraction has a fraction in its numerator or denominator. Simplify by multiplying numerator and denominator by the LCD of ALL inner fractions. Example: Simplify (1/x + 1/y)/(1/x - 1/y). LCD of inner fractions = xy. Multiply top and bottom by xy: Numerator: y + x. Denominator: y - x. Result: (x + y)/(y - x).

10. Solving Rational Equations

Multiply both sides by the LCD to clear all fractions. Solve the resulting polynomial equation. CHECK ALL SOLUTIONS — any value that makes a denominator zero is extraneous. Example: Solve 3/(x+2) + 1 = 7/(x+2). Multiply by (x+2): 3 + (x+2) = 7. x + 5 = 7 → x = 2. Check: 3/4 + 1 = 7/4? 3/4 + 4/4 = 7/4. True. x = 2 works.

11. Extraneous Solutions and Work Problems

Extraneous solution example: Solve x/(x-2) = 2/(x-2) + 3. Multiply by (x-2): x = 2 + 3(x-2) = 2 + 3x - 6 = 3x - 4. -2x = -4 → x = 2. Check: x = 2 makes (x-2) = 0. Denominator is zero. EXTRANEOUS. No solution. Work Problem: Pipe A fills a tank in 6 hours. Pipe B fills it in 4 hours. How long together? 1/6 + 1/4 = 1/t. LCD = 12: 2/12 + 3/12 = 5/12 = 1/t. t = 12/5 = 2.4 hours.

12. Common Mistakes

- Canceling terms instead of factors: (x^2 + 4)/(x + 4) cannot be simplified to x. - Forgetting to check for extraneous solutions after solving rational equations. - When subtracting fractions, not distributing the minus sign to all terms in the numerator. - Confusing division (flip and multiply) with multiplication. - Adding denominators: 1/3 + 1/5 ≠ 2/8. Always find the LCD first.