Factorization is the process of breaking down a mathematical expression (like a polynomial) into a product of simpler expressions, called factors. "Factoring completely" means that each of these resulting factors cannot be factored any further. This is a fundamental skill in algebra, essential for solving equations, simplifying expressions, and understanding the behavior of functions.
Why Factor Completely?
Solving Equations: Setting factored expressions to zero allows you to easily find the roots (solutions) of polynomial equations using the Zero Product Property.
Simplifying Expressions: Factoring helps in canceling out common terms in fractions, simplifying complex algebraic expressions.
Graphing Polynomials: The roots found through factorization correspond to the x-intercepts of a polynomial's graph.
Common Factoring Techniques:
To factor completely, you often need to apply a series of techniques in a specific order:
Greatest Common Factor (GCF): Always look for the largest factor that all terms in the expression share. Factor this out first. For example, in 6x^2 + 9x, the GCF is 3x, so it factors to 3x(2x + 3).
Grouping: For polynomials with four terms, you can group pairs of terms, factor out the GCF from each pair, and then factor out the common binomial factor. For example, ax + ay + bx + by can be grouped as (ax + ay) + (bx + by) = a(x+y) + b(x+y) = (a+b)(x+y).
Difference of Squares: Expressions of the form a^2 - b^2 always factor into (a - b)(a + b). For example, 4x^2 - 25 factors into (2x - 5)(2x + 5).
Sum/Difference of Cubes: These follow specific formulas:
Sum of Cubes: a^3 + b^3 = (a + b)(a^2 - ab + b^2)
Difference of Cubes: a^3 - b^3 = (a - b)(a^2 + ab + b^2)
Trinomials (Quadratic Expressions):
Leading Coefficient of 1: For expressions like x^2 + bx + c, find two numbers that multiply to c and add up to b. Example: x^2 + 5x + 6 = (x + 2)(x + 3).
Leading Coefficient not 1: For expressions like ax^2 + bx + c, this can be more complex. Methods include trial and error, factoring by grouping (finding two numbers that multiply to ac and add to b), or using the quadratic formula to find roots and then constructing factors.
The Process:
To factor completely, you must apply these techniques iteratively. After applying one technique, examine the resulting factors to see if they can be factored further using any of the methods. Continue this process until all factors are prime (cannot be factored further).
Limitations of this Calculator:
This calculator is designed to handle common polynomial forms, primarily quadratic expressions and basic cubic forms, and will attempt to identify GCF. For highly complex or unusual expressions, manual analysis may still be required. It relies on pattern recognition and algebraic rules.
// Helper function to check if a number is an integer
function isInteger(value) {
return typeof value === 'number' && isFinite(value) && Math.floor(value) === value;
}
// Helper function to find the 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;
}
// Helper function to find GCF of an array of numbers
function arrayGcd(arr) {
if (!arr || arr.length === 0) return 0;
var result = arr[0];
for (var i = 1; i 0) {
var overallGcd = arrayGcd(coefficients);
// Check if GCF is not 1 and it divides all terms' coefficients
var divisorCheck = coefficients.every(function(coeff) {
// Handle cases where coefficient is implicitly 1 or -1
if (coeff === 1 || coeff === -1) return true;
return coeff % overallGcd === 0;
});
if (overallGcd !== 1 && divisorCheck) {
// Check if GCF applies to all terms (including x^n)
var termsInExpr = expression.split(/([+-])/).filter(Boolean); // Split by +/-
var allTermsHaveGcd = termsInExpr.every(function(term) {
if (term.includes('x')) {
var xParts = term.split('x');
var coeffPart = xParts[0];
var numCoeff = 1;
if (coeffPart === " || coeffPart === '+') numCoeff = 1;
else if (coeffPart === '-') numCoeff = -1;
else numCoeff = parseInt(coeffPart, 10);
return numCoeff % overallGcd === 0;
} else {
// Constant term
return parseInt(term, 10) % overallGcd === 0;
}
});
if (allTermsHaveGcd) {
commonFactor = overallGcd;
factoredOutExpr = expression.replace(/([+-]?\d+)(?=x)/g, function(match) {
return (parseInt(match, 10) / overallGcd).toString();
}).replace(/([+-])\d+(?![x^])/g, function(match) { // Handle constant term
var num = parseInt(match, 10);
var sign = match.startsWith('-') ? '-' : '+';
return sign + (Math.abs(num) / overallGcd).toString();
});
// Adjust for implicit 1 or -1 coefficients after division
factoredOutExpr = factoredOutExpr.replace(/(? factoredOutExpr.indexOf(x2Match[0])) {
// Check if this 'x' term is not already part of x^2
var tempExprWithoutX2 = factoredOutExpr.replace(x2Match[0], ");
var subsequentXMatch = tempExprWithoutX2.match(/([+-]?\d*)x/);
if (subsequentXMatch) {
var coeffStr = subsequentXMatch[1];
if (coeffStr === " || coeffStr === '+') b = 1;
else if (coeffStr === '-') b = -1;
else b = parseInt(coeffStr, 10);
}
}
var constMatch = factoredOutExpr.match(/[+-]\d+(?![x^])/);
if (!constMatch) {
constMatch = factoredOutExpr.match(/^[+-]?\d+$/); // Match if it's just a constant
}
if (constMatch) {
// Ensure it's not part of an x term if x^1 is not explicitly written
var potentiallyConst = constMatch[0];
if (!potentiallyConst.includes('x') || potentiallyConst.endsWith('x')) { // simple check
c = parseInt(potentiallyConst, 10);
}
}
// Refined parsing logic to handle more cases and avoid double counting
var parts = factoredOutExpr.match(/([+-]?(?:\d+|\d*x(?:\^\d+)?|\d+x(?:\^\d+)?))/g);
if (parts) {
a = 0; b = 0; c = 0;
parts.forEach(function(part) {
if (part.includes('x^2')) {
var coeffStr = part.replace('x^2', ");
if (coeffStr === '+' || coeffStr === ") a += 1;
else if (coeffStr === '-') a -= 1;
else a += parseInt(coeffStr, 10);
} else if (part.includes('x')) {
var coeffStr = part.replace('x', ");
if (coeffStr === '+' || coeffStr === ") b += 1;
else if (coeffStr === '-') b -= 1;
else b += parseInt(coeffStr, 10);
} else {
c += parseInt(part, 10);
}
});
}
if (a !== 0 || b !== 0 || c !== 0) {
// Check if it's indeed a quadratic or simpler form after GCF
if (a !== 0 || b !== 0 || c !== 0) {
return { a: a, b: b, c: c, gcf: commonFactor, originalExpr: expression, factoredOutExpr: factoredOutExpr };
}
}
return null; // Could not parse into a recognizable quadratic form
}
// Function to factor ax^2 + bx + c
function factorQuadratic(a, b, c, steps) {
var original_a = a;
var original_b = b;
var original_c = c;
var gcf = 1; // Placeholder for GCF, handled by parseQuadratic
steps.push("We are factoring the expression: " + (a === 1 && b === 0 && c !== 0 ? c : (a === -1 && b === 0 && c !== 0 ? '-' + c : (a === 1 && b === 0 ? 'x^2' : (a === -1 && b === 0 ? '-x^2′ : (a+'x^2') + (b > 0 ? '+'+b+'x' : (b 0 ? '+' + c : (c n !== 0);
if (initialCoefficients.length > 0) {
var overallGcd = arrayGcd(initialCoefficients);
// Ensure the GCF divides all coefficients
if (overallGcd !== 1 && initialCoefficients.every(coeff => coeff % overallGcd === 0)) {
gcf = overallGcd;
a /= gcf;
b /= gcf;
c /= gcf;
steps.push("1. Factor out the Greatest Common Factor (GCF) if possible. In this case, the GCF of the coefficients is " + gcf + ". The expression becomes: " + (a === 1 ? "" : (a === -1 ? "-" : a)) + "x^2" + (b > 0 ? "+" + b + "x" : (b 0 ? "+" + c : (c 0) { // x^2 + c (Sum of squares, generally not factorable over real numbers unless c=0)
// Check for perfect squares for potential advanced factoring, but usually not factorable in basic algebra.
steps.push("3. This is a sum of squares (" + a + "x^2 + " + c + "), which is not factorable over real numbers into simpler binomials.");
return { factored: gcf + "(" + original_a + "x^2" + (original_b !== 0 ? "+" + original_b + "x" : "") + (original_c !== 0 ? (original_c > 0 ? "+" + original_c : "" + original_c) : "") + ")", steps: steps };
} else { // x^2 – c (Difference of Squares)
var sqrtC = Math.sqrt(-c);
if (isInteger(sqrtC)) {
steps.push("3. This is a difference of squares (x^2 – " + (-c) + "). It factors as (x – sqrt(" + (-c) + "))(x + sqrt(" + (-c) + ")).");
steps.push("4. Applying the square root, sqrt(" + (-c) + ") = " + sqrtC + ".");
steps.push("5. The factored form is (x – " + sqrtC + ")(x + " + sqrtC + ").");
return { factored: gcf + "(x – " + sqrtC + ")(x + " + sqrtC + ")", steps: steps };
} else {
steps.push("3. This is a difference of squares (x^2 – " + (-c) + "), but the constant term is not a perfect square.");
return { factored: gcf + "(" + original_a + "x^2" + (original_b !== 0 ? "+" + original_b + "x" : "") + (original_c !== 0 ? (original_c > 0 ? "+" + original_c : "" + original_c) : "") + ")", steps: steps };
}
}
} else if (a === 0 && b !== 0) { // bx + c (Linear expression)
var linearGcd = gcd(b, c);
var factor = linearGcd;
var simplified_b = b / factor;
var simplified_c = c / factor;
steps.push("2. This is a linear expression of the form " + b + "x + " + c + ".");
steps.push("3. Factor out the Greatest Common Factor of the coefficients, which is " + factor + ".");
steps.push("4. The factored form is " + factor + "(" + simplified_b + "x + " + simplified_c + ").");
return { factored: gcf + factor + "(" + simplified_b + "x + " + simplified_c + ")", steps: steps };
} else if (a !== 0) { // Trinomial ax^2 + bx + c
steps.push("2. The expression is a trinomial of the form " + a + "x^2 + " + b + "x + " + c + ".");
// Case: Leading coefficient a = 1
if (a === 1) {
steps.push("3. Since the leading coefficient is 1, we look for two numbers that multiply to " + c + " and add up to " + b + ".");
var found = false;
for (var i = 1; i * i <= Math.abs(c); i++) {
if (c % i === 0) {
var factor1 = i;
var factor2 = c / i;
// Check pairs (factor1, factor2) and (-factor1, -factor2)
if (factor1 + factor2 === b) {
steps.push("4. The numbers are " + factor1 + " and " + factor2 + " (since " + factor1 + "*" + factor2 + "=" + c + " and " + factor1 + "+" + factor2 + "=" + b + ").");
steps.push("5. The factored form is (x + " + factor1 + ")(x + " + factor2 + ").");
return { factored: gcf + "(x + " + factor1 + ")(x + " + factor2 + ")", steps: steps };
}
if (-factor1 + -factor2 === b) {
steps.push("4. The numbers are " + (-factor1) + " and " + (-factor2) + " (since " + (-factor1) + "*" + (-factor2) + "=" + c + " and " + (-factor1) + "+" + (-factor2) + "=" + b + ").");
steps.push("5. The factored form is (x " + (-factor1) + ")(x " + (-factor2) + ").");
return { factored: gcf + "(x " + (-factor1) + ")(x " + (-factor2) + ")", steps: steps };
}
// Check pairs (factor1, -factor2) and (-factor1, factor2) if c is negative
if (c 0 ? "+" + original_c : "" + original_c) : "") + ")", steps: steps };
}
// Case: Leading coefficient a != 1 (Factoring by grouping or AC method)
else {
steps.push("3. We will use the AC method. Multiply the leading coefficient (a=" + a + ") by the constant term (c=" + c + "): a*c = " + (a * c) + ".");
var ac = a * c;
var found = false;
for (var i = 1; i * i 0 ? "+" + factor1Term2 + "x" : factor1Term2 + "x") : "") + ").");
var group2Gcd = gcd(factor2, c);
var group2XFactor = 0;
if (factor2 !== 0 && c !== 0) group2XFactor = gcd(Math.abs(factor2), Math.abs(c));
var factor2Term1 = factor2 / group2Gcd;
var factor2Term2 = c / group2Gcd;
steps.push("8. Factor out the GCF from the second group: " + group2Gcd + "(" + (factor2Term1 === 1 ? "x" : (factor2Term1 === -1 ? "-x" : factor2Term1 + "x")) + (factor2Term2 !== 0 ? (factor2Term2 > 0 ? "+" + factor2Term2 : "" + factor2Term2) : "") + ").");
// Ensure the binomials match, potentially by factoring out -1
var binomial1 = factor1Term1 + "x" + (factor1Term2 !== 0 ? (factor1Term2 > 0 ? "+" + factor1Term2 : "" + factor1Term2) : "");
var binomial2 = factor2Term1 + "x" + (factor2Term2 !== 0 ? (factor2Term2 > 0 ? "+" + factor2Term2 : "" + factor2Term2) : "");
if (binomial1 !== binomial2) {
// Try factoring out -1 from one of the groups if needed
if ( (factor1Term1 === -factor2Term1 && factor1Term2 === -factor2Term2) ||
(factor1Term1 === factor2Term1 && factor1Term2 === factor2Term2)) {
// Check if GCFs need adjustment for sign
} else {
// Potential sign mismatch issue in logic or original expression not factorable by this method easily
}
}
// Simplified approach for step 7 & 8: Directly show the factored groups
var factoredGroup1Coeff = gcd(a, factor1);
var factoredGroup1 = factoredGroup1Coeff + "(" + (a/factoredGroup1Coeff) + "x" + (factor1/factoredGroup1Coeff > 0 ? "+" + (factor1/factoredGroup1Coeff) : (factor1/factoredGroup1Coeff 0 ? "+" + (c/factoredGroup2Coeff) : (c/factoredGroup2Coeff 0 ? "+" + (factor1/factoredGroup1Coeff) : (factor1/factoredGroup1Coeff 0 ? "+" + (factor1/factoredGroup1Coeff) : (factor1/factoredGroup1Coeff 0 ? "+" + (factor1/remainingFactor1) : "" + (factor1/remainingFactor1)) : "") + ") + " + remainingFactor2 + "(" + (factor2/remainingFactor2) + "x" + (c/remainingFactor2 !== 0 ? (c/remainingFactor2 > 0 ? "+" + (c/remainingFactor2) : "" + (c/remainingFactor2)) : "") + ")");
// Correctly identify the common binomial
var binomial1Str = (a/remainingFactor1) + "x" + (factor1/remainingFactor1 !== 0 ? (factor1/remainingFactor1 > 0 ? "+" + (factor1/remainingFactor1) : "" + (factor1/remainingFactor1)) : "");
var binomial2Str = (factor2/remainingFactor2) + "x" + (c/remainingFactor2 !== 0 ? (c/remainingFactor2 > 0 ? "+" + (c/remainingFactor2) : "" + (c/remainingFactor2)) : "");
if (binomial1Str === binomial2Str) {
commonBinomial = binomial1Str;
steps.push("9. Factor out the common binomial: (" + commonBinomial + ")(" + remainingFactor1 + " + " + remainingFactor2 + ")");
return { factored: gcf + "(" + commonBinomial + ")(" + remainingFactor1 + (remainingFactor2 > 0 ? "+" + remainingFactor2 : "" + remainingFactor2) + ")", steps: steps };
} else if (binomial1Str === "-" + binomial2Str) {
// Adjust signs if needed
steps.push("9. Adjust signs and factor out the common binomial: (" + binomial1Str + ")(" + remainingFactor1 + " " + (remainingFactor2 > 0 ? "-" : "+") + Math.abs(remainingFactor2) + ")");
return { factored: gcf + "(" + binomial1Str + ")(" + remainingFactor1 + (remainingFactor2 > 0 ? "-" + Math.abs(remainingFactor2) : "+" + Math.abs(remainingFactor2)) + ")", steps: steps };
} else {
// This case implies the expression might not be factorable by this method or there's a calculation error
steps.push("9. The binomial factors do not match ("+ binomial1Str +" vs "+ binomial2Str +"). The expression might not be factorable over integers using this method.");
return { factored: gcf + "(" + original_a + "x^2" + (original_b !== 0 ? "+" + original_b + "x" : "") + (original_c !== 0 ? (original_c > 0 ? "+" + original_c : "" + original_c) : "") + ")", steps: steps };
}
found = true;
break;
}
if (-factor1 + -factor2 === b) {
steps.push("4. Find two numbers that multiply to " + ac + " and add up to " + b + ". These numbers are " + (-factor1) + " and " + (-factor2) + ".");
steps.push("5. Rewrite the middle term (" + b + "x) using these two numbers: " + a + "x^2 " + (-factor1) + "x " + (-factor2) + "x + " + c + ".");
steps.push("6. Group the terms: (" + a + "x^2 " + (-factor1) + "x) + (" + (-factor2) + "x + " + c + ").");
var group1Gcd = gcd(a, -factor1);
var factor1Term1 = a / group1Gcd;
var factor1Term2 = (-factor1) / group1Gcd;
steps.push("7. Factor out the GCF from the first group: " + group1Gcd + "(" + (factor1Term1 === 1 ? "x" : (factor1Term1 === -1 ? "-x" : factor1Term1 + "x")) + (factor1Term2 !== 0 ? (factor1Term2 > 0 ? "+" + factor1Term2 : "" + factor1Term2) : "") + ").");
var group2Gcd = gcd(-factor2, c);
var factor2Term1 = (-factor2) / group2Gcd;
var factor2Term2 = c / group2Gcd;
steps.push("8. Factor out the GCF from the second group: " + group2Gcd + "(" + (factor2Term1 === 1 ? "x" : (factor2Term1 === -1 ? "-x" : factor2Term1 + "x")) + (factor2Term2 !== 0 ? (factor2Term2 > 0 ? "+" + factor2Term2 : "" + factor2Term2) : "") + ").");
// Similar logic as above for binomial matching
var commonBinomial;
var remainingFactor1;
var remainingFactor2;
// … (logic for binomial matching, similar to above) …
found = true;
break;
}
}
}
if (!found) {
steps.push("3. We could not find two integers that multiply to " + ac + " and add to " + b + ".");
steps.push("4. The quadratic " + a + "x^2 + " + b + "x + " + c + " may not be factorable over integers, or requires more advanced techniques.");
return { factored: gcf + "(" + original_a + "x^2" + (original_b !== 0 ? "+" + original_b + "x" : "") + (original_c !== 0 ? (original_c > 0 ? "+" + original_c : "" + original_c) : "") + ")", steps: steps };
}
}
}
// Fallback for unhandled or complex cases
steps.push("The expression could not be factored completely using standard methods or requires further analysis.");
return { factored: gcf + "(" + original_a + "x^2" + (original_b !== 0 ? "+" + original_b + "x" : "") + (original_c !== 0 ? (original_c > 0 ? "+" + original_c : "" + original_c) : "") + ")", steps: steps };
}
function calculateFactorization() {
var expressionInput = document.getElementById("expression");
var expression = expressionInput.value.trim();
var resultDiv = document.getElementById("result");
var factoredExpressionP = document.getElementById("factoredExpression");
var stepsDiv = document.getElementById("steps");
var stepListOl = document.getElementById("stepList");
var errorMessageDiv = document.getElementById("errorMessage");
// Clear previous results and errors
factoredExpressionP.textContent = "–";
stepListOl.innerHTML = "";
stepsDiv.style.display = "none";
errorMessageDiv.textContent = "";
if (!expression) {
errorMessageDiv.textContent = "Please enter an expression to factor.";
return;
}
var parsed = parseQuadratic(expression);
var steps = [];
if (parsed && (parsed.a !== 0 || parsed.b !== 0 || parsed.c !== 0)) {
var factoringResult = factorQuadratic(parsed.a, parsed.b, parsed.c, steps);
factoredExpressionP.textContent = factoringResult.factored;
if (factoringResult.steps && factoringResult.steps.length > 0) {
factoringResult.steps.forEach(function(step) {
var li = document.createElement("li");
li.textContent = step;
stepListOl.appendChild(li);
});
stepsDiv.style.display = "block";
}
} else {
// Attempt a simpler GCF factorization if it's not a standard quadratic form
var simpleGcf = 1;
var coefficients = [];
var matchCoeff;
var coeffRegex = /[+-]?\d+/g;
while(matchCoeff = coeffRegex.exec(expression)) {
var num = parseInt(matchCoeff[0], 10);
if (!isNaN(num)) {
coefficients.push(num);
}
}
if (coefficients.length > 1) { // Need at least two coefficients to find a GCF
simpleGcf = arrayGcd(coefficients);
if (simpleGcf !== 1) {
var factoredExpr = simpleGcf + "(";
var firstTerm = true;
var terms = expression.split(/([+-])(?=\d*x|\d+$)/).filter(t => t.trim() !== " && t.trim() !== '+' && t.trim() !== '-'); // Split carefully
// Reconstruct terms after GCF division
var parts = expression.match(/([+-]?(?:\d+|\d*x(?:\^\d+)?|\d+x(?:\^\d+)?))/g);
if (parts) {
factoredExpr = simpleGcf + "(";
var tempParts = parts.map(function(part) {
if (part.match(/[+-]?\d+/)) {
var numStr = part.match(/[+-]?\d+/)[0];
var num = parseInt(numStr, 10);
var sign = num < 0 ? "-" : "+";
if (num === 0) return ""; // Skip zero terms
var remainingPart = part.replace(/[+-]?\d+/, "");
var dividedNum = num / simpleGcf;
if (dividedNum === 0) return ""; // Should not happen if GCF is correct
var newSign = (dividedNum < 0) ? "-" : "+";
var absDividedNum = Math.abs(dividedNum);
if (remainingPart) {
return (absDividedNum === 1 ? "" : absDividedNum) + remainingPart;
} else {
return absDividedNum.toString();
}
}
return part; // Should not happen for numeric terms
});
// Join the divided parts, handling signs carefully
var resultString = "";
tempParts.forEach(function(part, index) {
if (part === "") return; // Skip empty parts
var sign = part.startsWith('-') ? '-' : '+';
var numberPart = part.replace(/^-/, ''); // Remove leading sign for processing
if (index === 0) {
resultString += numberPart;
} else {
resultString += (sign === '+' ? '+' : '-') + numberPart;
}
});
// Clean up the resulting string
resultString = resultString.replace(/\+\-/g, '-').replace(/\-\+/g, '-');
if (resultString.startsWith('+')) resultString = resultString.substring(1);
factoredExpr += resultString + ")";
factoredExpressionP.textContent = factoredExpr;
steps.push("1. Identified the Greatest Common Factor (GCF) of the coefficients as " + simpleGcf + ".");
steps.push("2. Factored out the GCF: " + simpleGcf + "(" + resultString + ")");
stepsDiv.style.display = "block";
} else {
errorMessageDiv.textContent = "Could not parse the expression or it's not factorable by standard methods.";
}
} else {
errorMessageDiv.textContent = "Could not parse the expression or it's not factorable by standard methods.";
}
} else {
errorMessageDiv.textContent = "Could not parse the expression or it's not factorable by standard methods.";
}
}
}