Simplify Rational Expression Calculator
Simplify rational expressions by factoring and canceling common factors with complete domain restrictions
Use ^ for exponents (e.g., x^2)
Enter polynomial expression
How to Simplify Rational Expressions
Step 1: Factor Completely
Factor both the numerator and denominator into their simplest forms. Look for:
- Greatest Common Factor (GCF)
- Difference of squares: a² - b² = (a+b)(a-b)
- Trinomials: ax² + bx + c
- Sum/difference of cubes
Step 2: Identify Restrictions
Set the original denominator equal to zero and solve. These values must be excluded from the domain.
Step 3: Cancel Common Factors
Cancel any factors that appear in both numerator and denominator. Remember: you can only cancel factors, not terms!
Step 4: Write Final Answer
Write the simplified expression with all domain restrictions clearly stated.
Frequently Asked Questions
Why must I state restrictions even after simplifying?
Restrictions are based on the original denominator. Even if a factor cancels during simplification, the values that made it zero are still excluded from the domain of the original expression.
Can I cancel terms instead of factors?
No! You can only cancel factors (expressions connected by multiplication). You cannot cancel terms (expressions connected by addition or subtraction). For example, in (x+2)/(x+3), you cannot cancel the x's.
What if the numerator and denominator share no common factors?
The expression is already in simplest form. You still need to identify and state the domain restrictions.
How do I factor trinomials?
For ax² + bx + c, find two numbers that multiply to ac and add to b. Use grouping or factoring patterns like (x + p)(x + q) where p·q = c and p + q = b.
What are the most common factoring patterns?
Difference of squares (a²-b²), perfect square trinomials (a²±2ab+b²), sum/difference of cubes, and standard trinomial factoring.
Should I multiply out my answer?
No! Leave the answer in factored form. This shows the cancellation process clearly and is generally considered the simplest form.
What if the denominator is a constant?
If the denominator is a non-zero constant, there are no restrictions. The expression is defined for all real numbers.
Can a rational expression have infinitely many restrictions?
No, polynomial denominators can only have a finite number of zeros, so there can only be finitely many restrictions.