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Rational Function Simplifier

Simplify (ax+b)/(cx+d), factor, cancel, and find domain

Enter rational expression (ax+b)/(cx+d)

R(x) = (x + ) / (x +)

Rational Expression Simplification

1. Factor numerator and denominator

2. Cancel common factors (both sides)

3. Check for holes (canceled factors)

4. State domain restrictions

Simplifying a rational expression requires factoring both numerator and denominator, canceling common factors, and identifying domain restrictions. Any canceled common factor creates a hole in the original function.

⚠ When you cancel a factor, the domain still excludes that x value from the original expression. The simplified form is equivalent everywhere the original is defined.

What Is Rational Expression Simplification?

A rational expression is a fraction with variables. To simplify: factor the numerator and denominator, cancel common factors. The result is the simplified expression in lowest terms. Canceled factors indicate holes in the graph.

Factorization

Factor both numerator and denominator. Look for common binomial factors like (x+k). Factor out constants first.

Cancellation

If numerator and denominator share a factor, cancel it. The canceled factor creates a hole at x = -k where k is the root of that factor.

Domain

Original domain excludes denominator zeros. Even after cancellation, those points are still excluded from the original expression.

Lowest Terms

The simplified form has no common factors. It is equivalent to the original wherever both are defined.

Teaching Example: (2x-4)/(3x-6). Factor num: 2(x-2). Factor den: 3(x-2). Cancel (x-2): 2/3. Hole at x=2. Domain: x!=2. Simplified: 2/3 for x!=2.

Applications

Algebra Calculus Limits Graphing Equation Solving Integration

Frequently Asked Questions

How to simplify rational?▼

Factor num and den, cancel common factors. Check for factors that are negatives of each other.

What is a hole?▼

A removable discontinuity. Occurs where a factor cancels. The simplified function is defined there, but the original is not.

Domain restrictions?▼

Exclude all x where original denominator=0. Canceled factors still exclude those x from the original function.

Lowest terms check?▼

After canceling all common factors, the expression is in lowest terms. No further simplification is possible.

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