// Function to calculate the Greatest Common Divisor (GCD) of two numbers
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 parse a single polynomial term string
// Expected format: [sign][coefficient][variable][^power]
// Examples: "6x^2", "12x", "-18", "x^3", "-y", "5"
function parseTerm(termString) {
termString = termString.trim();
if (termString === ") return null; // Empty input
var coeff = 1;
var variable = ";
var power = 0;
// Regex to match: [sign][coefficient][variable][^power]
// Group 1: Optional sign and digits for coefficient (e.g., "", "-", "6", "-12")
// Group 2: Optional single letter variable (e.g., "x", "y")
// Group 3: Optional power part (e.g., "^2")
// Group 4: The power digit itself (e.g., "2")
var match = termString.match(/^(-?\d*)?([a-zA-Z])?(\^(\d+))?$/);
if (!match) {
// If the main regex fails, try to match just a number (e.g., "10")
var numMatch = termString.match(/^(-?\d+)$/);
if (numMatch) {
coeff = parseInt(numMatch[1]);
return { coeff: coeff, variable: ", power: 0, original: termString };
}
// If still no match, it's an invalid format
return null;
}
// Parse coefficient
var coeffStr = match[1];
if (coeffStr === '-') {
coeff = -1;
} else if (coeffStr === ") {
// If no number is explicitly given, it's 1 (e.g., "x", "x^2″)
coeff = 1;
} else {
coeff = parseInt(coeffStr);
}
// Parse variable
variable = match[2] || ";
// Parse power
if (variable) { // If there's a variable
power = match[4] ? parseInt(match[4]) : 1; // If no explicit power, it's 1 (e.g., "x")
} else { // If no variable (constant term)
power = 0;
}
return { coeff: coeff, variable: variable, power: power, original: termString };
}
function calculateGCFFactor() {
var term1String = document.getElementById('term1Input').value;
var term2String = document.getElementById('term2Input').value;
var term3String = document.getElementById('term3Input').value;
var errorMessagesDiv = document.getElementById('errorMessages');
var gcfResultDiv = document.getElementById('gcfResult');
var factoredFormResultDiv = document.getElementById('factoredFormResult');
errorMessagesDiv.innerHTML = ";
gcfResultDiv.innerHTML = ";
factoredFormResultDiv.innerHTML = ";
var terms = [term1String, term2String, term3String];
var parsedTerms = [];
for (var i = 0; i < terms.length; i++) {
var parsed = parseTerm(terms[i]);
if (parsed) {
parsedTerms.push(parsed);
} else if (terms[i].trim() !== '') {
errorMessagesDiv.innerHTML += 'Error: Invalid format for term ' + (i + 1) + '. Please use format like "6x^2", "12x", "-18".';
}
}
if (parsedTerms.length < 2) {
errorMessagesDiv.innerHTML = 'Please enter at least two valid terms.';
return;
}
if (errorMessagesDiv.innerHTML !== '') {
return; // Stop if there are parsing errors
}
// — Calculate GCF of Coefficients —
var gcfCoeff = parsedTerms[0].coeff;
for (var i = 1; i < parsedTerms.length; i++) {
gcfCoeff = gcd(gcfCoeff, parsedTerms[i].coeff);
}
// — Determine Common Variable and its Minimum Power —
var commonVariable = '';
var minPower = Infinity;
// Check if all terms have a variable and if it's the same variable
var firstVariable = parsedTerms[0].variable;
var allTermsHaveVariable = true;
var allSameVariable = true;
if (firstVariable === '') { // If the first term is a constant
allTermsHaveVariable = false;
} else {
minPower = parsedTerms[0].power; // Initialize minPower with the first term's power
}
for (var i = 1; i 0)) {
gcfString += gcfCoeff;
}
if (commonVariable && minPower > 0) {
gcfString += commonVariable;
if (minPower > 1) {
gcfString += '^' + minPower;
}
}
if (gcfString === " || gcfString === '-') { // If GCF is just 1 or -1 and no variable part
if (gcfCoeff === -1 && (commonVariable && minPower > 0)) {
gcfString = '-'; // For -x^2
} else if (gcfCoeff === -1 && (!commonVariable || minPower === 0)) {
gcfString = '-1'; // For -1
} else {
gcfString = '1'; // For 1
}
}
// — Construct Remaining Polynomial —
var remainingTermsFormatted = [];
for (var i = 0; i 0)) {
termStr = "; // For 'x' instead of '1x'
} else if (newCoeff === -1 && (newVariable && newPower > 0)) {
termStr = '-'; // For '-x' instead of '-1x'
} else {
termStr += newCoeff;
}
if (newVariable && newPower > 0) {
termStr += newVariable;
if (newPower > 1) {
termStr += '^' + newPower;
}
} else if (newVariable && newPower === 0) {
// Variable was factored out completely, e.g., x^2 / x^2 = 1
// The coefficient already handles this.
}
// If it's a constant term (no variable), termStr is just newCoeff.
remainingTermsFormatted.push(termStr);
}
var remainingPolyString = remainingTermsFormatted[0];
for (var i = 1; i < remainingTermsFormatted.length; i++) {
var currentTerm = remainingTermsFormatted[i];
if (currentTerm.startsWith('-')) {
remainingPolyString += ' ' + currentTerm;
} else {
remainingPolyString += ' + ' + currentTerm;
}
}
// — Display Results —
gcfResultDiv.innerHTML = 'Greatest Common Factor (GCF): ' + gcfString;
factoredFormResultDiv.innerHTML = 'Factored Form: ' + gcfString + '(' + remainingPolyString + ')';
}
.calculator-container {
background-color: #f9f9f9;
border: 1px solid #ddd;
border-radius: 8px;
padding: 20px;
max-width: 600px;
margin: 20px auto;
font-family: Arial, sans-serif;
}
.calculator-container h2 {
text-align: center;
color: #333;
margin-bottom: 20px;
}
.calculator-content {
display: flex;
flex-direction: column;
}
.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;
}
.calculate-button {
background-color: #007bff;
color: white;
padding: 12px 20px;
border: none;
border-radius: 4px;
cursor: pointer;
font-size: 18px;
margin-top: 10px;
transition: background-color 0.3s ease;
}
.calculate-button:hover {
background-color: #0056b3;
}
.result-area {
margin-top: 25px;
padding: 15px;
background-color: #e9ecef;
border: 1px solid #dee2e6;
border-radius: 5px;
}
.result-area h3 {
color: #333;
margin-top: 0;
margin-bottom: 10px;
}
.result-output {
margin-bottom: 10px;
font-size: 1.1em;
color: #333;
word-wrap: break-word;
}
.result-output strong {
color: #007bff;
}
.error-message {
color: #dc3545;
font-weight: bold;
margin-top: 10px;
}
Understanding Factoring Using the Greatest Common Factor (GCF)
Factoring a polynomial is the process of breaking it down into a product of simpler expressions. One of the most fundamental methods of factoring is by using the Greatest Common Factor (GCF). This technique is essential for simplifying expressions, solving equations, and understanding more advanced algebraic concepts.
What is the Greatest Common Factor (GCF)?
The GCF of two or more numbers is the largest number that divides evenly into all of them. When dealing with polynomial terms, the GCF extends to include both the numerical coefficients and the variable parts.
Finding the GCF of Numbers:
To find the GCF of a set of numbers, you can list their factors or use prime factorization. For example, to find the GCF of 12 and 18:
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
The common factors are 1, 2, 3, and 6. The greatest among them is 6. So, GCF(12, 18) = 6.
Finding the GCF of Variables:
For variables, the GCF is the variable raised to the lowest power that appears in all terms. For example, to find the GCF of x^3, x^5, and x^2:
x^3 = x * x * x
x^5 = x * x * x * x * x
x^2 = x * x
The lowest power of x common to all terms is x^2. So, GCF(x^3, x^5, x^2) = x^2.
If terms have different variables (e.g., x and y) or a variable is not present in all terms, then that variable is not part of the GCF.
Steps to Factor a Polynomial Using GCF:
Identify the terms: Separate the polynomial into its individual terms.
Find the GCF of the coefficients: Determine the greatest common factor of all the numerical coefficients.
Find the GCF of the variables: For each variable that is common to ALL terms, find the lowest power it is raised to. If a variable is not in every term, it is not part of the GCF.
Combine the GCFs: Multiply the GCF of the coefficients by the GCF of the variables to get the overall GCF of the polynomial.
Divide each term by the GCF: Divide each original term of the polynomial by the combined GCF.
Write the factored form: Write the GCF outside a set of parentheses, and inside the parentheses, write the results of the division from step 5, connected by their original signs.
Example: Factoring 6x^2 + 12x
Terms:6x^2 and 12x.
GCF of coefficients: GCF(6, 12) = 6.
GCF of variables: Both terms have x. The powers are x^2 and x^1. The lowest power is x^1 (or just x).
Combined GCF:6 * x = 6x.
Divide each term by GCF:
6x^2 / 6x = x
12x / 6x = 2
Factored form:6x(x + 2).
Using the GCF Factoring Calculator
Our calculator simplifies this process for you. Simply enter up to three terms of your polynomial in the specified format (e.g., 6x^2, -12y, 5). The calculator will automatically determine the GCF and present the polynomial in its factored form.
Input Format Notes:
Enter terms like 6x^2 for 6 times x squared.
For terms with a variable but no explicit power, like 12x, it's assumed to be x^1.
For constant terms, just enter the number, like -18 or 5.
For terms like x^3 or -y, the coefficient is assumed to be 1 or -1 respectively.
This tool is perfect for students, educators, and anyone needing to quickly factor polynomials using the GCF method.