Rational expressions are fractions whose numerators and denominators are polynomials. They appear throughout algebra, precalculus, physics formulas, and rate problems because they describe quantities being divided or compared. Learning to operate with them helps you simplify complex formulas and solve equations more reliably.
The key idea is that rational expressions follow the same fraction rules you already know, but factoring and restrictions matter much more.
Key Facts
- A rational expression is undefined when its denominator equals 0.
- To simplify, factor first, then cancel common factors: (x^2 - 9)/(x^2 - 3x) = (x - 3)(x + 3)/(x(x - 3)) = (x + 3)/x, x ≠ 0, 3.
- To multiply, factor and cancel before multiplying: (a/b)(c/d) = ac/bd.
- To divide, multiply by the reciprocal: (a/b) ÷ (c/d) = (a/b)(d/c), where b ≠ 0, c ≠ 0, d ≠ 0.
- To add or subtract, use a common denominator: a/b + c/d = (ad + bc)/bd.
- Excluded values come from the original denominators, not only the simplified expression.
Vocabulary
- Rational expression
- A rational expression is a fraction with polynomials in the numerator, denominator, or both.
- Excluded value
- An excluded value is a value of the variable that makes an original denominator equal to zero.
- Common denominator
- A common denominator is a shared denominator used to add or subtract rational expressions.
- Factor
- A factor is an expression that is multiplied by another expression to make a product.
- Reciprocal
- The reciprocal of a fraction is found by switching its numerator and denominator.
Common Mistakes to Avoid
- Canceling terms instead of factors is wrong because only factors connected by multiplication can be canceled. In (x + 3)/x, the x in the denominator cannot cancel with part of x + 3.
- Forgetting excluded values is wrong because simplification can hide values that were not allowed in the original expression. Always find restrictions before canceling.
- Adding denominators is wrong because fractions do not add by combining denominators. Use a common denominator and add only the adjusted numerators.
- Dividing without flipping the second fraction is wrong because division by a fraction means multiplication by its reciprocal. Rewrite the operation before factoring and canceling.
Practice Questions
- 1 Simplify (x^2 - 16)/(x^2 + 2x - 8) and state all excluded values.
- 2 Multiply and simplify: (3x^2 - 12x)/(x^2 - 9) times (x + 3)/(6x). State all excluded values.
- 3 Explain why (x^2 - 4)/(x - 2) simplifies to x + 2 but is not exactly the same as x + 2 for every value of x.