The Greatest Common Factor (GCF) of two or more numbers is the largest number that divides evenly into each of them. When dealing with algebraic expressions, the GCF extends to include common variables raised to their lowest shared power. Factoring out the GCF is a fundamental technique in algebra used to simplify expressions, solve equations, and prepare polynomials for further factoring methods.
Why Factor Out the GCF?
Factoring out the GCF is crucial for several reasons:
Simplification: It makes complex expressions easier to understand and work with.
Solving Equations: If an equation is set to zero, factoring allows you to use the Zero Product Property to find solutions.
Further Factoring: It's often the first step in factoring more complex polynomials (e.g., trinomials, difference of squares).
Identifying Common Structures: It helps reveal underlying patterns in mathematical expressions.
How to Find the GCF of a Polynomial
To find and factor out the GCF from a polynomial, follow these steps:
Identify the Terms: Separate the polynomial into its individual terms. For example, in 12x^2 + 18x - 6xy, the terms are 12x^2, 18x, and -6xy.
Find the GCF of the Coefficients: Determine the greatest common factor of the numerical coefficients of all terms.
For 12, 18, -6, the absolute values are 12, 18, 6.
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
Factors of 6: 1, 2, 3, 6
The GCF of the coefficients is 6.
Find the GCF of the Variable Parts: For each variable present in all terms, take the variable raised to the lowest power it appears with.
In 12x^2, 18x, -6xy:
Variable 'x' appears as x^2, x^1, x^1. The lowest power is x^1 (or just x).
Variable 'y' appears as y^1 only in -6xy. It is not present in 12x^2 or 18x. Therefore, 'y' is not part of the GCF.
The GCF of the variable parts is x.
Combine to Form the Overall GCF: Multiply the GCF of the coefficients by the GCF of the variable parts.
Overall GCF = 6 * x = 6x.
Divide Each Term by the GCF: Divide each original term in the polynomial by the overall GCF.
12x^2 / 6x = 2x
18x / 6x = 3
-6xy / 6x = -y
Write the Factored Expression: Place the overall GCF outside a set of parentheses, and write the results of the division inside the parentheses, connected by their original signs.
Factored expression: 6x(2x + 3 - y)
Examples
Let's look at a few more examples:
Example 1: Factor 15y^3 - 25y^2 + 5y
Coefficients: 15, -25, 5. GCF = 5.
Variables: y^3, y^2, y^1. Lowest power of y is y^1.
Variables: 'a' and 'b' are not common to both terms.
Overall GCF = 1.
Since the GCF is 1, the expression cannot be factored further by GCF.
Use the calculator above to practice factoring out the GCF from various polynomial expressions!
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font-family: 'Arial', sans-serif;
background-color: #f9f9f9;
border: 1px solid #ddd;
border-radius: 8px;
padding: 20px;
max-width: 800px;
margin: 20px auto;
box-shadow: 0 2px 4px rgba(0,0,0,0.1);
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.calculator-container h2 {
color: #333;
margin-bottom: 20px;
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.calculator-content {
background-color: #ffffff;
padding: 20px;
border-radius: 5px;
border: 1px solid #eee;
margin-bottom: 20px;
}
.input-group {
margin-bottom: 15px;
}
.input-group label {
display: block;
margin-bottom: 5px;
color: #555;
font-weight: bold;
}
.input-group input[type="text"] {
width: calc(100% – 22px);
padding: 10px;
border: 1px solid #ccc;
border-radius: 4px;
font-size: 16px;
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.calculate-button {
background-color: #007bff;
color: white;
padding: 12px 20px;
border: none;
border-radius: 4px;
cursor: pointer;
font-size: 16px;
width: 100%;
box-sizing: border-box;
transition: background-color 0.2s;
}
.calculate-button:hover {
background-color: #0056b3;
}
.result-area {
margin-top: 20px;
padding: 15px;
border: 1px solid #e0e0e0;
border-radius: 4px;
background-color: #f0f0f0;
min-height: 50px;
color: #333;
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margin: 5px 0;
font-size: 1.1em;
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color: #007bff;
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margin-bottom: 10px;
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margin-bottom: 10px;
margin-left: 20px;
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.article-content ul li, .article-content ol li {
margin-bottom: 5px;
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// Function to parse a polynomial string into an array of term objects
function parsePolynomial(polyString) {
polyString = polyString.replace(/\s+/g, "); // Remove all spaces
if (polyString === ") return [];
// Ensure leading sign for first term if not present (e.g., "12x" becomes "+12x")
if (polyString[0] !== '-' && polyString[0] !== '+') {
polyString = '+' + polyString;
}
var terms = [];
// Split by '+' and '-' while keeping the signs for individual term parsing
// Example: "+12x^2+18x+-6y" -> ["+12x^2", "+18x", "-6y"]
var parts = polyString.replace(/-/g, '+-').split('+').filter(Boolean);
for (var i = 0; i < parts.length; i++) {
var termStr = parts[i];
var currentCoeff = 1;
var currentVars = {};
var remaining = termStr;
// Check for leading sign and coefficient
var coeffMatch = remaining.match(/^([-+]?\d*\.?\d*)/);
if (coeffMatch && coeffMatch[1] !== '') {
var coeffPart = coeffMatch[1];
if (coeffPart === '-' || coeffPart === '+') {
currentCoeff = parseInt(coeffPart + '1');
} else {
currentCoeff = parseFloat(coeffPart);
}
remaining = remaining.substring(coeffPart.length);
} else if (remaining.startsWith('-')) { // e.g., -x
currentCoeff = -1;
remaining = remaining.substring(1);
} else if (remaining.startsWith('+')) { // e.g., +x (shouldn't happen if we split by '+')
currentCoeff = 1;
remaining = remaining.substring(1);
} else { // e.g., x (no explicit coeff)
currentCoeff = 1;
}
// Parse variable parts (e.g., x^2y)
var varMatches = remaining.matchAll(/([a-zA-Z])(?:\^(\d+))?/g);
for (var vMatch of varMatches) {
var variable = vMatch[1];
var exponent = vMatch[2] ? parseInt(vMatch[2]) : 1;
currentVars[variable] = exponent;
}
terms.push({ coefficient: currentCoeff, variables: currentVars });
}
return terms;
}
// Euclidean algorithm for Greatest Common Divisor (GCD)
function gcd(a, b) {
a = Math.abs(a);
b = Math.abs(b);
while (b) {
var temp = b;
b = a % b;
a = temp;
}
return a;
}
// Function to get the GCF of an array of coefficients
function getGCFCoefficients(coeffs) {
if (coeffs.length === 0) return 1;
var result = coeffs[0];
for (var i = 1; i < coeffs.length; i++) {
result = gcd(result, coeffs[i]);
}
return result;
}
// Function to get the GCF of variable parts across multiple terms
function getGCFVariables(terms) {
if (terms.length === 0) return {};
if (terms.length === 1) return terms[0].variables;
var commonVars = {};
var firstTermVars = terms[0].variables;
// Initialize commonVars with variables from the first term
for (var varName in firstTermVars) {
if (firstTermVars.hasOwnProperty(varName)) {
commonVars[varName] = firstTermVars[varName];
}
}
// Iterate through remaining terms to find common variables and min exponents
for (var i = 1; i "x^2y")
function formatVariablePart(vars) {
var varStr = ";
var sortedVars = Object.keys(vars).sort(); // Consistent order
for (var i = 0; i 1) {
varStr += varName + '^' + exponent;
}
// If exponent is 0, it's not included
}
return varStr;
}
// Function to format a coefficient and variable part into a term string (e.g., 12, {x:2} -> "12x^2″)
function formatTerm(coeff, vars) {
var varPart = formatVariablePart(vars);
if (Object.keys(vars).length === 0) { // Pure number term
return coeff.toString();
}
if (coeff === 1) {
return varPart;
}
if (coeff === -1) {
return '-' + varPart;
}
return coeff.toString() + varPart;
}
// Main calculation function
function calculateGCF() {
var polynomialInput = document.getElementById('polynomialInput').value;
var resultDiv = document.getElementById('result');
resultDiv.innerHTML = ";
if (polynomialInput.trim() === ") {
resultDiv.innerHTML = 'Please enter a polynomial expression.';
return;
}
var terms;
try {
terms = parsePolynomial(polynomialInput);
if (terms.length === 0) {
resultDiv.innerHTML = 'Could not parse the polynomial. Please check the format.';
return;
}
} catch (e) {
resultDiv.innerHTML = 'Error parsing polynomial: ' + e.message + ";
return;
}
if (terms.length === 1) {
resultDiv.innerHTML = 'The expression has only one term: ' + formatTerm(terms[0].coefficient, terms[0].variables) + '. No factoring needed.';
return;
}
// Extract coefficients for GCF calculation
var coeffs = terms.map(function(term) { return term.coefficient; });
var gcfCoeff = getGCFCoefficients(coeffs);
// Convention: if the first term's coefficient is negative, factor out a negative GCF
if (terms.length > 0 && terms[0].coefficient < 0) {
gcfCoeff = -Math.abs(gcfCoeff);
} else {
gcfCoeff = Math.abs(gcfCoeff);
}
var gcfVars = getGCFVariables(terms);
// Construct the GCF string for display
var gcfStringRaw = formatTerm(gcfCoeff, gcfVars);
// Handle special cases for GCF = 1 or -1
if (gcfStringRaw === '1' && Object.keys(gcfVars).length === 0) {
resultDiv.innerHTML = 'The Greatest Common Factor (GCF) is 1. The polynomial cannot be factored further by GCF.';
resultDiv.innerHTML += 'Original Expression: ' + polynomialInput + '';
return;
}
var actualGcfCoeff = gcfCoeff; // Use this for division
if (gcfStringRaw === '-1' && Object.keys(gcfVars).length === 0) {
gcfStringRaw = '-'; // Display as '-'
actualGcfCoeff = -1; // Ensure division by -1
}
// Divide each term by the GCF
var remainingTerms = [];
for (var i = 0; i < terms.length; i++) {
var term = terms[i];
var newCoeff = term.coefficient / actualGcfCoeff;
var newVars = {};
for (var varName in term.variables) {
if (term.variables.hasOwnProperty(varName)) {
var gcfVarExp = gcfVars[varName] || 0;
var newExp = term.variables[varName] – gcfVarExp;
if (newExp 0) {
newVars[varName] = newExp;
}
}
}
remainingTerms.push({ coefficient: newCoeff, variables: newVars });
}
// Format the remaining polynomial
var remainingPolyString = ";
for (var i = 0; i 0 && rTerm.coefficient > 0) {
remainingPolyString += '+';
}
remainingPolyString += formatted;
}
// Final output
var finalResult = gcfStringRaw + '(' + remainingPolyString + ')';
resultDiv.innerHTML = 'The Greatest Common Factor (GCF) is: ' + gcfStringRaw + '';
resultDiv.innerHTML += 'Factored Expression: ' + finalResult + '';
}