Rational Expressions & Equations

A rational expression is a fraction whose numerator and denominator are polynomials. Working with them is essentially working with fractions -- but with variables. The same rules you learned for numeric fractions (common denominators, canceling common factors) apply here, with one crucial addition: you must watch for values that make the denominator zero.

Simplifying Rational Expressions

The strategy: factor, then cancel common factors.

Worked Example 1 -- Simplifying

Simplify: (x² - 9) / (x² + 5x + 6).

Factor numerator: x² - 9 = (x - 3)(x + 3).
Factor denominator: x² + 5x + 6 = (x + 2)(x + 3).
Cancel the common factor (x + 3): result is (x - 3)/(x + 2), with x ≠ -3, x ≠ -2.

Multiplying and Dividing

Multiplying: factor everything, cancel common factors across all numerators and denominators, then multiply what remains.

Dividing: multiply by the reciprocal of the divisor, then proceed as above.

Worked Example 2 -- Multiplying

Multiply: (x² - 4)/(x + 1) · (x + 1)/(x + 2).

Factor: x² - 4 = (x - 2)(x + 2).
Rewrite: [(x - 2)(x + 2)]/(x + 1) · (x + 1)/(x + 2).
Cancel (x + 1) and (x + 2): result is (x - 2).

Adding and Subtracting

Just like numeric fractions, you need a common denominator.

Factor each denominator completely.
Find the LCD (least common denominator): include each factor the maximum number of times it appears.
Rewrite each fraction with the LCD.
Combine numerators; simplify.

Worked Example 3 -- Adding

Add: 3/(x - 1) + 2/(x + 4).

LCD = (x - 1)(x + 4).
Rewrite: 3(x + 4)/[(x - 1)(x + 4)] + 2(x - 1)/[(x - 1)(x + 4)].
Combine: [3(x + 4) + 2(x - 1)] / [(x - 1)(x + 4)] = (3x + 12 + 2x - 2) / [(x - 1)(x + 4)].
Simplify numerator: (5x + 10) / [(x - 1)(x + 4)] = 5(x + 2) / [(x - 1)(x + 4)].

Solving Rational Equations

To solve an equation involving rational expressions:

Find the LCD of all denominators in the equation.
Multiply every term on both sides by the LCD to clear all fractions.
Solve the resulting polynomial equation.
Check for extraneous solutions -- any solution that makes a denominator zero must be rejected.

Extraneous Solutions -- Do Not Skip This Step

When you multiply both sides by an expression containing a variable, you may introduce solutions that are not valid. Always substitute your answers back into the original equation and verify that no denominator equals zero.

Worked Example 4 -- Solving a Rational Equation

Solve: x/(x - 2) = 4/(x - 2) + 1.

LCD = (x - 2). Multiply every term by (x - 2).
x = 4 + (x - 2).
x = 4 + x - 2 → x = x + 2 → 0 = 2.
This is a contradiction -- there is no solution.

Note: if we had gotten x = 2, we would reject it because x = 2 makes the denominator zero.

Factoring First Saves Time

Always factor every denominator before finding the LCD. This often reveals common factors that make the LCD smaller and the algebra simpler.

Practice Problems

Simplify: (x² - x - 6) / (x² - 9).
Show Solution

Factor: (x - 3)(x + 2) / (x - 3)(x + 3). Cancel (x - 3): (x + 2)/(x + 3), x ≠ 3.
Divide: (x² - 1)/(x + 3) ÷ (x - 1)/(x + 3).
Show Solution

Multiply by reciprocal: [(x - 1)(x + 1)/(x + 3)] · [(x + 3)/(x - 1)]. Cancel (x + 3) and (x - 1): result is x + 1.
Add: 5/(x + 2) + 3/(x - 1).
Show Solution

LCD = (x + 2)(x - 1). Result: [5(x - 1) + 3(x + 2)] / [(x + 2)(x - 1)] = (8x + 1)/[(x + 2)(x - 1)].
Solve: 2/x + 3/(x + 1) = 1.
Show Solution

LCD = x(x + 1). Multiply: 2(x + 1) + 3x = x(x + 1). 2x + 2 + 3x = x² + x. So x² - 4x - 2 = 0. x = (4 ± √24)/2 = 2 ± √6. Both values are valid (neither makes a denominator zero). x ≈ 4.449 or x ≈ -0.449.

Summary

Factor and cancel to simplify rational expressions.
Multiply straight across; divide by flipping the second fraction.
Add and subtract by finding a common denominator.
Solve rational equations by clearing denominators, then always check for extraneous solutions.