Enter your rational expression in the form of a fraction (Numerator / Denominator). This calculator will attempt to simplify it by factoring and canceling common terms.
Understanding Rational Algebraic Expressions
A rational algebraic expression is essentially a fraction where the numerator and the denominator are polynomials. These expressions are fundamental in algebra and appear frequently in calculus, engineering, and physics. The general form of a rational expression is:
P(x) / Q(x)
where P(x) and Q(x) are polynomials, and Q(x) is not the zero polynomial.
Why Simplify?
Simplifying a rational expression means rewriting it in its simplest form by canceling out any common factors between the numerator and the denominator. This process is crucial for:
Easier analysis and evaluation of the expression.
Solving equations involving rational expressions.
Performing operations like addition, subtraction, multiplication, and division of rational expressions.
Understanding the behavior of functions, such as finding vertical asymptotes and holes.
How to Simplify (Manual Steps)
The general approach to simplifying rational expressions involves the following steps:
Factor the Numerator: Find all the factors of the polynomial in the numerator. This might involve techniques like factoring out a greatest common factor (GCF), difference of squares, sum/difference of cubes, or trinomial factoring.
Factor the Denominator: Similarly, completely factor the polynomial in the denominator.
Identify Common Factors: Look for identical factors that appear in both the factored numerator and the factored denominator.
Cancel Common Factors: Divide both the numerator and the denominator by each common factor. Remember that any factor you cancel must not be equal to zero.
Write the Simplified Expression: The remaining expression in the numerator divided by the remaining expression in the denominator is the simplified form.
Step 1 (Factor Numerator): Factor out the GCF, 2x: 2x(x + 2).
Step 2 (Factor Denominator): Factor the difference of squares: (x - 2)(x + 2).
Step 3 & 4 (Identify & Cancel Common Factors): The common factor is (x + 2). Canceling it leaves:
[2x(x + 2)] / [(x - 2)(x + 2)] = 2x / (x - 2)
Simplified Expression:2x / (x - 2) (where x ≠ 2 and x ≠ -2)
Important Note: This calculator uses a simplified approach and may not handle all complex polynomial factorizations or symbolic manipulations perfectly. For advanced cases, manual methods or specialized symbolic computation software are recommended.
// Basic factoring and simplification logic for polynomials
// This is a highly simplified implementation and will not handle all cases.
// It focuses on common binomials and simple trinomials.
function gcd(a, b) {
a = Math.abs(a);
b = Math.abs(b);
while (b) {
var t = b;
b = a % b;
a = t;
}
return a;
}
// Very basic factoring functions – a real symbolic math engine is required for full functionality.
// This function attempts to factor simple polynomials like x^2 – a^2, x^2 + bx + c, etc.
// It's highly limited.
function factorPolynomial(polyStr) {
polyStr = polyStr.trim().toLowerCase().replace(/\s+/g, "); // Clean input
if (polyStr === "") return { factors: [], isConstant: true, value: "" };
if (polyStr === "1") return { factors: ["1"], isConstant: true, value: "1" };
if (polyStr === "0") return { factors: ["0"], isConstant: true, value: "0" };
// Check for simple linear forms like "ax+b"
var linearMatch = polyStr.match(/^([+-]?\d*)x([+-]\d+)?$/);
if (linearMatch) {
var coeffX = linearMatch[1] === " || linearMatch[1] === '+' ? 1 : (linearMatch[1] === '-' ? -1 : parseInt(linearMatch[1]));
var constant = linearMatch[2] ? parseInt(linearMatch[2]) : 0;
if (coeffX === 1 && constant === 0) return { factors: ["x"], isConstant: false };
if (coeffX === -1 && constant === 0) return { factors: ["-x"], isConstant: false };
if (constant === 0) return { factors: [`${coeffX === 1 ? " : coeffX}x`], isConstant: false };
// Cannot easily factor ax+b into simple factors for this basic approach.
return { factors: [polyStr], isConstant: false };
}
// Check for simple quadratic forms like "x^2 – a^2" (difference of squares)
var diffOfSquaresMatch = polyStr.match(/^x\^2([+-])(\d+)$/);
if (diffOfSquaresMatch) {
var sign = diffOfSquaresMatch[1];
var num = parseInt(diffOfSquaresMatch[2]);
if (sign === '-') {
var root = Math.sqrt(num);
if (Number.isInteger(root)) {
return { factors: [`(x-${root})`, `(x+${root})`], isConstant: false };
}
}
}
// Check for simple quadratic forms like "x^2 + bx + c"
var trinomialMatch = polyStr.match(/^x\^2([+-]\d+)x([+-]\d+)$/);
if (trinomialMatch) {
var b = parseInt(trinomialMatch[1]);
var c = parseInt(trinomialMatch[2]);
var factors = [];
for (var i = 1; i 0) return { factors: factors, isConstant: false };
}
// Check for GCF in expressions like ax^2 + bx
var gcfLinearMatch = polyStr.match(/^([+-]?\d*)x\^2([+-]\d*)x$/);
if (gcfLinearMatch) {
var coeffX2 = gcfLinearMatch[1] === " || gcfLinearMatch[1] === '+' ? 1 : (gcfLinearMatch[1] === '-' ? -1 : parseInt(gcfLinearMatch[1]));
var coeffX = gcfLinearMatch[2] === " || gcfLinearMatch[2] === '+' ? 1 : (gcfLinearMatch[2] === '-' ? -1 : parseInt(gcfLinearMatch[2]));
if (coeffX === 0) coeffX = 1; // Handle case like 2x^2
if (coeffX2 === 0) coeffX2 = 1; // Handle case like 2x
var commonDivisor = gcd(coeffX2, coeffX);
var commonFactor = `${commonDivisor}x`;
var remainingPoly = `${coeffX2 / commonDivisor}x + ${coeffX / commonDivisor}`;
return { factors: [commonFactor, remainingPoly], isConstant: false };
}
// If no specific pattern matches, return the original string as a single factor.
return { factors: [polyStr], isConstant: false };
}
function simplifyExpression() {
var numeratorStr = document.getElementById('numerator').value;
var denominatorStr = document.getElementById('denominator').value;
var resultDiv = document.getElementById('result');
resultDiv.innerHTML = "; // Clear previous result
if (!numeratorStr || !denominatorStr) {
resultDiv.innerHTML = 'Please enter both a numerator and a denominator.';
return;
}
// Basic attempt to represent polynomials and factor them.
// This is a placeholder for a real symbolic math library.
// A full implementation would require parsing expressions,
// performing polynomial division, GCD of polynomials, etc.
var numFactorsResult = factorPolynomial(numeratorStr);
var denFactorsResult = factorPolynomial(denominatorStr);
var numFactors = numFactorsResult.factors;
var denFactors = denFactorsResult.factors;
var simplifiedNum = [];
var simplifiedDen = [];
var cancelledFactors = [];
// Simplified cancellation logic: check if any factor from num exists in den
var denFactorsCopy = […denFactors]; // Create a mutable copy
for (var i = 0; i < numFactors.length; i++) {
var currentNumFactor = numFactors[i];
var foundMatch = false;
for (var j = 0; j 0 ? simplifiedNum.join(") : "1";
var finalDenStr = simplifiedDen.length > 0 ? simplifiedDen.join(") : "1";
// Handle cases where numerator or denominator becomes effectively 1 or 0
if (finalNumStr === "" || finalNumStr === "1") finalNumStr = "1";
if (finalDenStr === "" || finalDenStr === "1") finalDenStr = "1";
if (finalNumStr === "0" && finalDenStr !== "0") {
finalNumStr = "0";
finalDenStr = "1";
} else if (finalDenStr === "0") {
resultDiv.innerHTML = 'Error: Denominator cannot be zero after simplification.';
return;
}
var simplifiedExpressionStr = `${finalNumStr} / ${finalDenStr}`;
var originalExpressionStr = `${numeratorStr} / ${denominatorStr}`;
if (cancelledFactors.length > 0) {
resultDiv.innerHTML = `Original: ${originalExpressionStr}Simplified: ${simplifiedExpressionStr}(Cancelled: ${cancelledFactors.join(', ')})`;
} else {
resultDiv.innerHTML = `Original: ${originalExpressionStr}Expression is already simplified or cannot be simplified further by this calculator.`;
}
}