ALGEBRAAlgebraMathematics Calculator
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Rational Expressions

P(x)/Q(x) — polynomial over polynomial. Simplify by canceling common factors. Domain: exclude roots of Q. Holes: canceled factors. Vertical asymptotes: remaining roots of denominator.

Concept Fundamentals
Cancel GCF
Simplify
a/b+c/d=(ad+bc)/bd
Add
(a/b)(c/d)=ac/bd
Multiply
Q(x)≠0
Domain

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Hole at x=a if (x−a) cancels in numerator and denominator. Vertical asymptote at x=a if (x−a) remains in denominator after simplifying. LCD = product of distinct factors from all denominators.

Key quantities
Cancel GCF
Simplify
Key relation
a/b+c/d=(ad+bc)/bd
Add
Key relation
(a/b)(c/d)=ac/bd
Multiply
Key relation
Q(x)≠0
Domain
Key relation

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Why: Rational expressions model rates, concentrations, and transfer functions. Simplifying reveals structure. Domain restrictions prevent division by zero. Holes vs asymptotes: canceled vs uncanceled roots.

How: Simplify: factor numerator and denominator, cancel common factors. Add/subtract: find LCD, convert, combine. Multiply: multiply across. Divide: multiply by reciprocal. Domain: set denominator ≠ 0.

Hole at x=a if (x−a) cancels in numerator and denominator.Vertical asymptote at x=a if (x−a) remains in denominator after simplifying.
Sources:Khan Academy — Rational ExpressionsWolfram MathWorld — Rational Function

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Simplify RationalAdd, subtract, multiply, divide

📐 Examples — Click to Load

Expression 1: P(x) / Q(x)

e.g. 1,0,-4 for x²-4
e.g. 1,-2 for x-2
rational.sh
CALCULATED
$ rational --simplify
Simplified
x+2
Domain: x ≠ 2
Holes: x = 2
Vertical asymptotes: none
Canceled: (x - 2)
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Coefficient Comparison

Canceled vs Remaining Factors

📐 Calculation Steps

Original: (x^2-4) / (x-2)
Common factors (cancel): (x - 2)
Holes at: x = 2
Vertical asymptotes: none
Simplified: x+2
Domain: x ≠ 2

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

📐

(x²−1)/(x−1)=x+1 for x≠1. Hole at x=1.

— Hole Example

⁄

1/x has vertical asymptote at x=0 — domain x≠0.

— Asymptote

📋 Domain Restrictions

A rational expression p(x)/q(x) is undefined when q(x)=0. Always exclude these values from the domain. Example: (x²-4)/(x-2) has domain x≠2 because x-2=0 when x=2.

  • • Holes: Values that make both numerator and denominator zero. After canceling, the factor is gone but the value is still excluded.
  • • Vertical asymptotes: Values that make only the denominator zero (after cancellation). The graph approaches ±∞.
  • • Simplification: Factor both numerator and denominator, cancel common factors. The simplified form equals the original for all x in the domain.

🔄 Operations with Fractions

➕Addition: Find LCD, convert each fraction, add numerators.Source: Add
➖Subtraction: Same as addition but subtract numerators.Source: Subtract
✖️Multiplication: Multiply numerators, multiply denominators.Source: Multiply
➗Division: Multiply by the reciprocal of the divisor.Source: Divide

📖 Simplification Rules

Step 1: Factor the numerator and denominator completely.

Step 2: Identify common factors.

Step 3: Cancel common factors. The canceled values are holes in the graph.

Step 4: Remaining denominator roots are vertical asymptotes.

Example: (x²-4)/(x-2) = (x-2)(x+2)/(x-2) = x+2 for x≠2. Hole at x=2.

📊 Holes vs Asymptotes

TypeConditionGraph behavior
HoleNum and den both zeroRemovable discontinuity
VAOnly den zero (after cancel)Graph → ±∞

📐 Worked Examples

(x²-4)/(x-2)

Factor: x²-4 = (x-2)(x+2). Cancel (x-2): result = x+2 for x≠2. Hole at x=2.

(x²+5x+6)/(x²+3x+2)

Num = (x+2)(x+3), Den = (x+1)(x+2). Cancel (x+2): (x+3)/(x+1) for x≠-1,-2. Hole at x=-2, VA at x=-1.

Real-world: Work rate

If A does 1/x job/hr and B does 1/(x+1) job/hr, combined rate = 1/x + 1/(x+1) = (2x+1)/(x(x+1)).

⚠️ Input format: Enter coefficients as comma-separated, highest degree first. E.g. "1,0,-4" for x²-4, "1,-2" for x-2.

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