Enter the dividend and divisor polynomials below. Use 'x' for the variable and '^' for exponents (e.g., 3x^2 - 5x + 2). Ensure coefficients are numbers and exponents are non-negative integers.
Results:
Quotient:
Remainder:
function calculatePolynomialDivision() {
var dividendStr = document.getElementById('dividendPolynomial').value.trim();
var divisorStr = document.getElementById('divisorPolynomial').value.trim();
// Helper function to parse a polynomial string into a map of exponent -> coefficient
function parsePolynomial(polyStr) {
var terms = {};
polyStr = polyStr.replace(/\s+/g, "); // Remove all spaces
if (polyStr === ") {
return {0: 0}; // Represents the zero polynomial
}
// Normalize terms: ensure 'x' has '1' coefficient and 'x^1' exponent if missing
// e.g., +x -> +1x^1, -x -> -1x^1
polyStr = polyStr.replace(/([+-])x(?!\^)/g, '$11x^1');
// e.g., x -> 1x^1
polyStr = polyStr.replace(/^x(?!\^)/g, '1x^1');
// e.g., 2x -> 2x^1
polyStr = polyStr.replace(/([+-]?\d*\.?\d*)x(?!\^)/g, '$1x^1');
// Handle terms like 'x^2' (no explicit '1' coefficient)
// e.g., +x^2 -> +1x^2
polyStr = polyStr.replace(/([+-])x(\^\d+)/g, '$11x$2');
// e.g., x^2 -> 1x^2
polyStr = polyStr.replace(/^x(\^\d+)/g, '1x$1');
// Handle constant terms (e.g., "5", "-3")
// e.g., +5 -> +5x^0, -3 -> -3x^0
polyStr = polyStr.replace(/([+-]?\d*\.?\d+)$/, '$1x^0');
// If the whole string is just a number (e.g., "5")
if (polyStr.match(/^[+-]?\d*\.?\d+$/)) {
polyStr += 'x^0';
}
// Regex to find terms: [+-]? (optional sign) (\d*\.?\d*) (coefficient) x^(\d+) (variable and exponent)
var termRegex = /([+-]?\d*\.?\d*)x\^(\d+)/g;
var match;
while ((match = termRegex.exec(polyStr)) !== null) {
var coeffStr = match[1];
var expStr = match[2];
var coeff = parseFloat(coeffStr);
if (isNaN(coeff)) {
document.getElementById('quotientResult').innerHTML = 'Error: Invalid coefficient in polynomial.';
document.getElementById('remainderResult').innerHTML = ";
return null;
}
var exp = parseInt(expStr);
if (isNaN(exp) || exp coefficient to a sorted array [coeff_0, coeff_1, …, coeff_n]
function polyMapToArray(polyMap) {
var maxExp = 0;
for (var exp in polyMap) {
if (polyMap.hasOwnProperty(exp)) {
maxExp = Math.max(maxExp, parseInt(exp));
}
}
var coeffs = new Array(maxExp + 1).fill(0);
for (var exp in polyMap) {
if (polyMap.hasOwnProperty(exp)) {
coeffs[parseInt(exp)] = polyMap[exp];
}
}
// Remove leading zeros (highest power terms with 0 coefficient)
while (coeffs.length > 1 && coeffs[coeffs.length – 1] === 0) {
coeffs.pop();
}
return coeffs;
}
// Helper function to convert coefficient array back to polynomial string
function formatPolynomial(coeffs) {
if (coeffs.length === 0 || (coeffs.length === 1 && coeffs[0] === 0)) {
return '0';
}
var terms = [];
for (var i = coeffs.length – 1; i >= 0; i–) {
var coeff = coeffs[i];
if (coeff === 0) continue;
var term = ";
// Add sign for positive terms if not the first term
if (coeff > 0 && terms.length > 0) {
term += '+';
} else if (coeff 1) { // Exponent > 1
term += '^' + i;
}
}
return terms.join(");
}
// Helper function for polynomial subtraction (poly1 – poly2)
function polySubtract(poly1, poly2) {
var result = [];
var maxLength = Math.max(poly1.length, poly2.length);
for (var i = 0; i 1 && result[result.length – 1] === 0) {
result.pop();
}
return result;
}
// Helper function for polynomial multiplication by a term (poly * (coeff * x^exp))
function polyMultiplyTerm(poly, termCoeff, termExp) {
var result = new Array(poly.length + termExp).fill(0);
for (var i = 0; i = divisorDegree && remainder[remainder.length – 1] !== 0) {
var remainderDegree = remainder.length – 1;
var remainderLeadCoeff = remainder[remainderDegree];
// If remainder's leading coefficient is 0, but degree is still high, pop it.
// This can happen if a subtraction results in a 0 leading term.
while (remainder.length > 1 && remainder[remainder.length – 1] === 0) {
remainder.pop();
remainderDegree = remainder.length – 1;
if (remainderDegree < divisorDegree) break; // Remainder degree is now less than divisor degree
remainderLeadCoeff = remainder[remainderDegree];
}
if (remainderDegree < divisorDegree) break; // Exit if remainder degree is now less than divisor degree
// Calculate term for quotient
var termCoeff = remainderLeadCoeff / divisorLeadCoeff;
var termExp = remainderDegree – divisorDegree;
// Add term to quotient
// Ensure quotient array is large enough
while (quotient.length 1 && remainder[remainder.length – 1] === 0) {
remainder.pop();
}
}
return { quotient: quotient, remainder: remainder };
}
var dividendMap = parsePolynomial(dividendStr);
var divisorMap = parsePolynomial(divisorStr);
if (dividendMap === null || divisorMap === null) {
// Error message already set by parsePolynomial
return;
}
var dividendCoeffs = polyMapToArray(dividendMap);
var divisorCoeffs = polyMapToArray(divisorMap);
var result = performPolyDivision(dividendCoeffs, divisorCoeffs);
if (result.error) {
document.getElementById('quotientResult').innerHTML = 'Error: ' + result.error;
document.getElementById('remainderResult').innerHTML = ";
} else {
document.getElementById('quotientResult').innerHTML = formatPolynomial(result.quotient);
document.getElementById('remainderResult').innerHTML = formatPolynomial(result.remainder);
}
}
Understanding Polynomial Division
Polynomial division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree. It's a fundamental operation in algebra, often used to simplify rational expressions, find roots of polynomials, and factor polynomials. The process is analogous to the long division of numbers, yielding a quotient and a remainder.
The Division Algorithm for Polynomials
For any two polynomials, a dividend P(x) and a non-zero divisor D(x), there exist unique polynomials Q(x) (the quotient) and R(x) (the remainder) such that:
P(x) = D(x) * Q(x) + R(x)
where the degree of R(x) is less than the degree of D(x), or R(x) = 0.
Steps for Polynomial Long Division
Arrange Polynomials: Write both the dividend and the divisor in descending powers of the variable. If any powers are missing, include them with a coefficient of zero (e.g., x^3 + 0x^2 - 2x + 5).
Divide Leading Terms: Divide the leading term of the dividend by the leading term of the divisor. This gives the first term of the quotient.
Multiply: Multiply the entire divisor by this first term of the quotient.
Subtract: Subtract the result from the dividend. Be careful with signs! This gives a new polynomial.
Bring Down: Bring down the next term from the original dividend (if any) to form a new dividend.
Repeat: Repeat steps 2-5 with the new polynomial as the dividend until the degree of the remainder is less than the degree of the divisor.
Example:
Let's divide P(x) = x^3 - 2x^2 - 4 by D(x) = x - 3.
Setup:
________
x - 3 | x^3 - 2x^2 + 0x - 4
Step 1: Divide x^3 (leading term of dividend) by x (leading term of divisor) to get x^2. This is the first term of the quotient.
x^2
________
x - 3 | x^3 - 2x^2 + 0x - 4
Step 2: Multiply the divisor (x - 3) by x^2 to get x^3 - 3x^2.
The remainder is 5, which has a degree (0) less than the degree of the divisor (1). The division process stops here.
Result:
Quotient: x^2 + x + 3
Remainder: 5
Why Use This Calculator?
This calculator automates the tedious process of polynomial long division, providing instant results for the quotient and remainder. It's a valuable tool for students learning algebra, checking homework, or for anyone needing to quickly perform polynomial division without manual calculation errors. Simply input your dividend and divisor polynomials, and the calculator will handle the complex algebraic manipulations for you.