What Is a Rational Expression?
A rational expression is any expression that can be written as a ratio of two polynomials. Understanding rational expressions is prerequisite to working with rational functions, performing polynomial long division, applying partial fraction decomposition, and ultimately integrating rational functions in calculus.
Domain Restrictions (Excluded Values)
Simplifying Rational Expressions
The key to simplifying rational expressions is factoring both the numerator and denominator completely, then canceling common factors.
Multiplying and Dividing Rational Expressions
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Finding the Least Common Denominator (LCD)
Building Equivalent Fractions
Combining and Simplifying
Complex Fractions
Common Mistakes
Canceling terms instead of factors
You can only cancel common factors (things that are multiplied), not common terms (things that are added). For example, (x+3)/(x+5) cannot be simplified by canceling the x's. Only after factoring can you cancel.
Forgetting domain restrictions after canceling
When you cancel a factor like (x-2), the restriction x ≠ 2 still applies. The simplified expression looks different but must have the same domain as the original.
Sign errors when subtracting rational expressions
When subtracting, the negative sign must be distributed across the entire numerator of the second fraction. Forgetting to distribute leads to incorrect results.
Practice Problems
Related Guides
Frequently Asked Questions
What is a rational expression?
A rational expression is a fraction whose numerator and denominator are both polynomials. Examples include (x+1)/(x-2), 5/x, and (x²-9)/(x²-4). They follow the same rules as numeric fractions but require extra attention to domain restrictions (values that make the denominator zero).
How do you simplify a rational expression?
Factor both the numerator and the denominator completely, then cancel any common factors. For example, (x²-9)/(x²+5x+6) factors to (x+3)(x-3)/((x+2)(x+3)), and the (x+3) cancels to give (x-3)/(x+2). Always state the domain restrictions — x ≠ -3 and x ≠ -2 — even after canceling.
What are domain restrictions in rational expressions?
Domain restrictions are the values of the variable that make the denominator equal to zero. These values must be excluded because division by zero is undefined. You must find domain restrictions before simplifying, because canceling a factor hides the restriction but does not remove it.
How do you add or subtract rational expressions?
Find the least common denominator (LCD) of all the expressions, rewrite each fraction with the LCD as its denominator, then add or subtract the numerators. Finally, simplify the resulting expression by factoring and canceling if possible. The process mirrors adding numeric fractions like 1/3 + 1/4.
What is a complex fraction?
A complex fraction is a fraction that has fractions in its numerator, denominator, or both. For example, (1/x + 1)/(1 - 1/x) is a complex fraction. To simplify, multiply the numerator and denominator by the LCD of all the mini-fractions, which eliminates the nested fractions in one step.
How are rational expressions related to rational functions?
A rational function is simply a function defined by a rational expression: f(x) = P(x)/Q(x). The skills you use to simplify rational expressions — factoring, finding domain restrictions, combining fractions — are exactly the skills needed to analyze rational functions, find their asymptotes, and integrate them in calculus.
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