Rational Expressions Online Calculator

Calculator

Enter univariate polynomials in x. Supported examples include x^2 - 1, 3x^3 - 2x + 5, and 1.

Expression A numerator

Expression A denominator

Operation

Expression B numerator

Expression B denominator

Evaluate at x

Graph minimum x

Graph maximum x

Graph step

Decimal precision

Example Data Table

Case	A numerator	A denominator	B numerator	B denominator	Operation	Expected idea
1	x^2 - 1	x - 1	x + 2	x^2 - 4	Simplify	Cancel x - 1, but keep x ≠ 1.
2	x + 1	x - 2	2x	x + 3	Add	Use a common denominator.
3	x^2 - 9	x + 3	x - 4	1	Multiply	Cancel shared factors first or after multiplying.
4	x^2 + 5x + 6	x^2 - 4	x + 3	x - 2	Divide	Multiply by the reciprocal of B.

Formula Used

Let A = N₁ / D₁ and B = N₂ / D₂. Denominators must not be zero.

Simplify: divide numerator and denominator by their greatest common polynomial factor.
Add: A + B = (N₁D₂ + N₂D₁) / (D₁D₂).
Subtract: A − B = (N₁D₂ − N₂D₁) / (D₁D₂).
Multiply: A × B = (N₁N₂) / (D₁D₂).
Divide: A ÷ B = (N₁D₂) / (D₁N₂), where N₂ cannot be zero.
Evaluate: substitute x into the simplified expression, then divide numerator value by denominator value.

How to Use This Calculator

Type the numerator and denominator for expression A.
Add expression B when you want addition, subtraction, multiplication, or division.
Select the operation from the menu.
Enter an x value if you want a numerical check.
Adjust graph range and decimal precision if needed.
Press Calculate. The result appears above the form.
Use CSV or PDF download buttons to save the answer.

Understanding Rational Expressions

What They Mean

Rational expressions are fractions made from polynomials. They look like numeric fractions, but the numerator and denominator contain variables. This makes them powerful in algebra. They describe rates, ratios, curves, and many changing quantities. A rational expression is only valid when its denominator is not zero.

Why Simplifying Matters

Simplifying removes common polynomial factors. The expression becomes cleaner and easier to study. A canceled factor still matters for the domain. For example, (x² - 1) / (x - 1) simplifies to x + 1, but x cannot equal 1. The original denominator created that restriction. This is why the calculator shows both the simplified expression and the excluded values.

Combining Expressions

Addition and subtraction need a common denominator. Multiplication uses numerator times numerator and denominator times denominator. Division changes into multiplication by the reciprocal. These rules match ordinary fractions. The difference is that each part may contain several terms and powers of x. Careful combining helps prevent sign mistakes and missing factors.

Using Graphs and Tables

A graph helps reveal shape, intercepts, gaps, and vertical breaks. Tables show exact or approximate values at selected points. When a denominator becomes zero, the value is undefined. The calculator marks those cases instead of forcing a number. This protects the result from false conclusions. It also helps students compare algebraic work with visual behavior.

Common Mistakes

Many errors happen when users cancel terms instead of factors. A term is only part of a sum. A factor multiplies the whole expression. You may cancel common factors across the numerator and denominator. You should not cancel a single x from x + 2. Factoring first makes this rule easier to see.

Good Algebra Habits

Always check restrictions before accepting a simplified answer. Write the operation rule first. Combine like terms carefully. Factor when possible. Then cancel only common factors, not separate terms. Finally, test the result with a safe x value. These steps make rational expression work clearer, faster, and more reliable for homework, revision, teaching, and checking.

FAQs

1. What is a rational expression?

A rational expression is a fraction whose numerator and denominator are polynomials. The denominator cannot equal zero, so domain restrictions must be checked.

2. Can this calculator simplify polynomial fractions?

Yes. It finds common polynomial factors using division steps, cancels them, and displays the reduced form with original restrictions.

3. Why do restrictions remain after cancellation?

Restrictions come from the original expression. If a canceled denominator factor was zero at some x value, that value still cannot be used.

4. Which polynomial format should I type?

Use terms like x^2 - 5x + 6, 3x^3 + 2x - 1, or 1. Parentheses are not supported in direct inputs.

5. Can I add or subtract two rational expressions?

Yes. Enter expression A and expression B. The calculator builds a common denominator, combines numerators, and simplifies the final fraction.

6. What happens during division?

The second expression is inverted, then multiplied. The divisor expression must not be zero, so its numerator also creates restrictions.

7. Why is my value undefined?

A value is undefined when the denominator equals zero at the chosen x. Try another x value outside the restriction list.

8. What does the graph show?

The graph shows the simplified rational expression over your selected range. Points near undefined denominator values are separated to avoid misleading lines.