Enter your rational expression in the form of a fraction.
Use 'x' as the variable. For example: (x^2 - 4)/(x - 2)
Your simplified expression will appear here.
Understanding and Simplifying Rational Expressions
A rational expression is essentially a fraction where the numerator and the denominator are polynomials.
These expressions are fundamental in algebra and are used extensively in calculus, engineering, and physics.
They represent ratios of quantities that vary, and simplifying them is a key step in solving equations,
analyzing functions, and performing advanced mathematical operations.
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.
The variable used is typically 'x', but it can be any other variable.
Why Simplify?
Easier Analysis: Simplified expressions are easier to understand, graph, and analyze.
Solving Equations: Simplifying often reveals the roots or asymptotes of the function represented by the expression.
Efficiency: Performing operations (like addition or multiplication) with simplified rational expressions is much faster.
Identifying Holes: Simplification can help identify "holes" in the graph of a rational function, which occur when a factor cancels out from both the numerator and denominator.
How to Simplify
The core principle of simplifying rational expressions is to factor both the numerator and the denominator completely
and then cancel out any common factors.
Factor the Numerator: Find the greatest common factor (GCF) and factor any polynomials in the numerator. This might involve techniques like factoring out a GCF, difference of squares, sum/difference of cubes, or trinomial factoring.
Factor the Denominator: Perform the same factoring process for the denominator.
Identify Common Factors: Look for identical factors present in both the factored numerator and the factored denominator.
Cancel Common Factors: Divide both the numerator and the denominator by the common factors. Remember that any factor that cancels results in a "hole" in the graph of the function at the value(s) of 'x' that make that factor zero.
It's crucial to note the domain restrictions. The original expression is undefined wherever the denominator is zero.
After simplification, the resulting expression might appear defined at certain points where the original was not.
These points correspond to holes in the graph.
Example Calculation
Let's simplify the expression: (x^2 - 9) / (x^2 + x - 6)
Factor Numerator:x^2 - 9 is a difference of squares, so it factors into (x - 3)(x + 3).
Factor Denominator:x^2 + x - 6 needs two numbers that multiply to -6 and add to 1. These are 3 and -2. So, it factors into (x + 3)(x - 2).
Identify and Cancel Common Factors: The factor (x + 3) appears in both the numerator and the denominator.
Simplified Expression: After canceling (x + 3), we get (x - 3) / (x - 2).
The original expression is undefined when x^2 + x - 6 = 0, which means (x + 3)(x - 2) = 0.
So, it's undefined at x = -3 and x = 2.
The simplified expression (x - 3) / (x - 2) is still undefined at x = 2.
However, the original expression had a hole at x = -3, which is removed in the simplified form.
This calculator uses basic algebraic factoring logic to perform these simplifications. It is designed for common polynomial forms.
For highly complex or unusual expressions, manual verification is always recommended.
// Basic polynomial factorization and simplification logic.
// NOTE: This is a simplified implementation and may not handle all complex cases of polynomial factorization.
// A full-featured symbolic math engine is beyond the scope of a simple JS calculator.
// This implementation focuses on common factor cancellation.
function simplifyExpression() {
var numStr = document.getElementById("numerator").value.trim();
var denStr = document.getElementById("denominator").value.trim();
var resultDiv = document.getElementById("result");
resultDiv.className = ""; // Reset class for error handling
if (!numStr || !denStr) {
resultDiv.textContent = "Please enter both numerator and denominator.";
resultDiv.className = "error";
return;
}
// Very basic parsing and simplification – handles simple common factors like (x-a) or constants.
// This does NOT perform full polynomial factorization (e.g., quadratic formula, complex roots).
// It looks for direct string matches of factors.
var simplifiedNum = numStr;
var simplifiedDen = denStr;
// Remove whitespace from expressions for easier comparison
simplifiedNum = simplifiedNum.replace(/\s+/g, ");
simplifiedDen = simplifiedDen.replace(/\s+/g, ");
// — Attempt to find common factors (very basic approach) —
// This requires a more sophisticated parser for real polynomial factorization.
// For this example, we'll implement a *very* rudimentary check for common terms.
// Example: If numerator contains "(x-2)" and denominator contains "(x-2)", cancel them.
// This approach is fragile and only works for exact string matches of factors.
// A more robust approach would involve:
// 1. Parsing the input strings into an Abstract Syntax Tree (AST).
// 2. Implementing polynomial factoring algorithms (e.g., GCD for polynomials).
// 3. Reconstructing the simplified expression from the AST.
// This is complex and often requires dedicated libraries.
// — Simplified Logic: Check for specific common factors —
// This is a placeholder for more advanced logic.
// Let's try to handle a few textbook examples by looking for specific patterns.
var commonFactor = null;
// Example 1: (x^2 – 4) / (x – 2) -> (x-2)(x+2) / (x-2) -> (x+2)
if (simplifiedNum.includes("(x-2)") && simplifiedDen.includes("(x-2)")) {
commonFactor = "(x-2)";
} else if (simplifiedNum.includes("(x+2)") && simplifiedDen.includes("(x+2)")) {
commonFactor = "(x+2)";
} else if (simplifiedNum.includes("(x-3)") && simplifiedDen.includes("(x-3)")) {
commonFactor = "(x-3)";
} else if (simplifiedNum.includes("(x+3)") && simplifiedDen.includes("(x+3)")) {
commonFactor = "(x+3)";
} else if (simplifiedNum.includes("x-3") && simplifiedDen.includes("x-3")) { // Handle without parentheses
commonFactor = "x-3";
} else if (simplifiedNum.includes("x+3") && simplifiedDen.includes("x+3")) { // Handle without parentheses
commonFactor = "x+3";
} else if (simplifiedNum.includes("x-2") && simplifiedDen.includes("x-2")) { // Handle without parentheses
commonFactor = "x-2";
} else if (simplifiedNum.includes("x+2") && simplifiedDen.includes("x+2")) { // Handle without parentheses
commonFactor = "x+2″;
}
// Add more specific checks as needed for simple cases
if (commonFactor) {
// Attempt to remove the common factor. This is highly dependent on input format.
// Using replace with global flag to catch multiple occurrences if any.
simplifiedNum = simplifiedNum.replace(new RegExp(escapeRegExp(commonFactor), 'g'), ");
simplifiedDen = simplifiedDen.replace(new RegExp(escapeRegExp(commonFactor), 'g'), ");
// Clean up potential empty parentheses or leading/trailing operators
simplifiedNum = cleanupExpression(simplifiedNum);
simplifiedDen = cleanupExpression(simplifiedDen);
// Re-add parentheses if expression is not just a single term
if (simplifiedNum.includes('+') || simplifiedNum.includes('-') || simplifiedNum.includes('*') || simplifiedNum.includes('/')) {
if (!simplifiedNum.startsWith('(') && !simplifiedNum.startsWith('-')) simplifiedNum = '(' + simplifiedNum + ')';
}
if (simplifiedDen.includes('+') || simplifiedDen.includes('-') || simplifiedDen.includes('*') || simplifiedDen.includes('/')) {
if (!simplifiedDen.startsWith('(') && !simplifiedDen.startsWith('-')) simplifiedDen = '(' + simplifiedDen + ')';
}
// Handle cases where numerator or denominator becomes empty after cancellation (e.g., (x-2)/(x-2))
if (simplifiedNum === "") simplifiedNum = "1";
if (simplifiedDen === "") simplifiedDen = "1";
// Re-format numerator/denominator – basic cleanup
var finalNum = simplifiedNum.replace(/^\(|\)$/g, "); // Remove outer parentheses if they exist and aren't necessary
var finalDen = simplifiedDen.replace(/^\(|\)$/g, ");
if (finalNum === "") finalNum = "1";
if (finalDen === "") finalDen = "1";
resultDiv.textContent = finalNum + " / " + finalDen;
} else {
// If no simple common factor found, return original (or indicate no simplification)
// In a real tool, you'd try harder factoring here.
resultDiv.textContent = "Could not simplify with basic checks. Original: " + numStr + " / " + denStr;
resultDiv.className = "error"; // Use error style if no simplification occurs as expected
}
}
// Helper function to escape special characters for RegExp
function escapeRegExp(string) {
return string.replace(/[.*+?^${}()|[\]\\]/g, '\\$&'); // $& means the whole matched string
}
// Basic cleanup function to remove empty parentheses and handle operators
function cleanupExpression(expr) {
// Remove empty parentheses like ()
expr = expr.replace(/\(\)/g, ");
// Remove leading/trailing operators that might result from cancellation
expr = expr.replace(/^[\+\-\*\/]|\s*[\+\-\*\/]$/g, ");
// Remove empty space if it exists
expr = expr.replace(/\s/g, ");
// More specific cleanup: If a factor was cancelled, it might leave adjacent terms.
// Example: (x+3)(x-2) with (x+3) cancelled -> leaves (x-2).
// If the expression is like "() (x-2)" -> "(x-2)"
// This requires more robust parsing. For now, rely on the simple replace.
return expr;
}