Rational Expression Simplifier Calculator
Solve rational expression simplifier problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
P(x)/Q(x) simplified by factoring and canceling GCF
Factor both the numerator polynomial P(x) and denominator polynomial Q(x) completely, identify their greatest common factor (GCF), and divide both by the GCF to obtain the simplified form. Domain restrictions from the original denominator must be preserved.
Worked Examples
Example 1: Simplifying a Rational Expression with Common Factors
Problem:Simplify (x^2 - 5x + 6) / (x^2 - 4x + 3).
Solution:Factor numerator: x^2 - 5x + 6 = (x - 2)(x - 3)\nFactor denominator: x^2 - 4x + 3 = (x - 1)(x - 3)\nCommon factor: (x - 3)\nSimplified: (x - 2) / (x - 1)\nRestrictions: x cannot equal 3 (hole) or 1 (vertical asymptote)\nHole at x = 3: y = (3-2)/(3-1) = 1/2
Result:Simplified: (x - 2)/(x - 1) | Hole at (3, 0.5) | Asymptote at x = 1
Example 2: Rational Expression with No Common Factors
Problem:Simplify (x^2 + x - 6) / (x^2 + 5x + 6).
Solution:Factor numerator: x^2 + x - 6 = (x + 3)(x - 2)\nFactor denominator: x^2 + 5x + 6 = (x + 2)(x + 3)\nCommon factor: (x + 3)\nSimplified: (x - 2) / (x + 2)\nRestrictions: x cannot equal -3 (hole) or -2 (vertical asymptote)\nHole at x = -3: y = (-3-2)/(-3+2) = -5/(-1) = 5
Result:Simplified: (x - 2)/(x + 2) | Hole at (-3, 5) | Asymptote at x = -2
Frequently Asked Questions
What is a rational expression and how do you simplify one?
A rational expression is a fraction where both the numerator and denominator are polynomials, such as (x^2 - 4)/(x^2 - 4x + 4). To simplify a rational expression, you factor both the numerator and denominator completely, then cancel any common factors that appear in both. For example, (x^2 - 4)/(x^2 - 4x + 4) factors as (x-2)(x+2)/((x-2)(x-2)), and after canceling one (x-2) factor, the simplified form is (x+2)/(x-2). It is critical to note domain restrictions: the original expression is undefined at x = 2, so even after simplification, x = 2 must be excluded from the domain.
How do you multiply and divide rational expressions?
To multiply rational expressions, factor all numerators and denominators completely, then multiply numerators together and denominators together, canceling common factors across the entire expression. For example, (x+1)/(x-2) times (x-2)/(x+3) simplifies to (x+1)/(x+3) after canceling (x-2). To divide rational expressions, multiply by the reciprocal of the divisor. So (A/B) divided by (C/D) becomes (A/B) times (D/C) = AD/(BC). Always factor before multiplying to make cancellation easier. State all domain restrictions from the original expressions as well as any values that make a divisor zero.
How do you add and subtract rational expressions?
Adding and subtracting rational expressions requires a common denominator, similar to adding ordinary fractions. First, factor each denominator. Then find the least common denominator (LCD) by taking each factor to its highest power. Multiply each fraction by the appropriate form of 1 to obtain the LCD. Finally, add or subtract the numerators and simplify the result. For example, to add 1/(x-1) + 2/(x+1), the LCD is (x-1)(x+1). Rewrite as (x+1)/((x-1)(x+1)) + 2(x-1)/((x-1)(x+1)) = (x+1+2x-2)/((x-1)(x+1)) = (3x-1)/(x^2-1).
What are horizontal and oblique asymptotes of rational functions?
Horizontal asymptotes describe the behavior of a rational function as x approaches positive or negative infinity. If the numerator degree is less than the denominator degree, the horizontal asymptote is y = 0. If the degrees are equal, the horizontal asymptote is y = (leading coefficient of numerator)/(leading coefficient of denominator). If the numerator degree is exactly one more than the denominator degree, there is an oblique (slant) asymptote found by polynomial long division. If the numerator degree exceeds the denominator degree by two or more, there is no horizontal or oblique asymptote, and the function grows without bound.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy