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Algebra 2 Unit 4: Rational expressions & equations

Rational expressions are fractions of polynomials. Work with them like numerical fractions, and watch for values that make a denominator zero.

Covers: Simplifying, multiplying and dividing, adding with common denominators, solving rational equations, extraneous solutions

Study guide

Rational expressions are fractions of polynomials. To simplify, factor the numerator and denominator completely and cancel common factors (factors, never individual terms). Keep track of restricted values: any x that makes the original denominator zero is excluded, even if its factor cancels.

To multiply rational expressions, factor everything, cancel, and multiply straight across. To divide, multiply by the reciprocal of the divisor. To add or subtract, rewrite each fraction over a common denominator, ideally the least common denominator (LCD) found by factoring each denominator, then combine numerators and simplify. Distribute subtraction carefully across the whole second numerator.

To solve a rational equation, multiply both sides by the LCD to clear the fractions, then solve the resulting polynomial equation. Cross-multiplying works when the equation is a single fraction equal to another.

Because clearing fractions can introduce values that make an original denominator zero, you must check every answer. A result that fails the original equation is an extraneous solution and gets thrown out. Work and rate problems are common applications: rates add, so if one pipe fills a tank in 3 hours and another in 6, together they fill 1/3 + 1/6 = 1/2 of the tank per hour.

When a rational expression contains a sum or difference of cubes, factor it with the cube patterns before canceling. For example, (x³ + 8)/(x + 2) = x² − 2x + 4 for x ≠ −2. To simplify a complex fraction, multiply the numerator and denominator by the LCD of all the small fractions inside it, then simplify the result.

Key ideas

• Factor completely before canceling.
• Add or subtract with a common denominator.
• Divide by multiplying by the reciprocal.
• Solve rational equations by multiplying both sides by the LCD.
• Reject any solution that makes an original denominator zero.

Formulas and rules

Common mistakes

• Canceling terms that aren't factors.
• Keeping an extraneous solution.

Words to know

Simplifying rational expressions

Factor top and bottom completely, then cancel common factors.

Dividing rational expressions

Multiply by the reciprocal of the divisor.

First step to solve a rational equation

Multiply both sides by the least common denominator.

Extraneous solution

A result that fails the original equation (e.g. makes a denominator zero).

Restricted values

x-values that make a denominator 0; still excluded after canceling.

Rational expression

A fraction whose numerator and denominator are polynomials.

Least common denominator (LCD)

The smallest expression every denominator divides into.

Multiplying rational expressions

Factor, cancel, then multiply across.

Complex fraction

A fraction containing fractions; multiply top and bottom by the LCD.

Combined work rate

Rates add: 1/3 + 1/6 = 1/2 of the job per hour.

Sample practice questions

17 more questions like these, from level 1 to 5, are in the practice set.