Dividing Polynomials Long Division Calculator Step by Step

Polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree. It's similar to the familiar long division process for numbers, but applied to algebraic expressions. This method is crucial for factoring polynomials, finding roots, and simplifying rational expressions.

How Polynomial Long Division Works

The process involves a series of steps:

  1. Set up the division: Write the dividend and divisor in descending powers of the variable. If any powers are missing, include them with a coefficient of zero (e.g., x^3 + 5 becomes x^3 + 0x^2 + 0x + 5).
  2. Divide the leading terms: Divide the first term of the dividend by the first term of the divisor. This gives you the first term of the quotient.
  3. Multiply: Multiply the term you just found in the quotient by the entire divisor.
  4. Subtract: Subtract the result from the dividend. Be careful with signs!
  5. Bring down: Bring down the next term from the original dividend.
  6. Repeat: Continue steps 2-5 with the new polynomial (the result of the subtraction plus the brought-down term) until the degree of the remainder is less than the degree of the divisor.

The final result is expressed as: Dividend / Divisor = Quotient + Remainder / Divisor.

Using the Polynomial Long Division Calculator

This calculator helps you perform polynomial long division step-by-step. To use it:

  1. Enter Dividend Coefficients: Input the coefficients of your dividend polynomial, separated by commas. Start with the coefficient of the highest degree term. If a term is missing (e.g., no x^2 term in x^3 - 2x + 5), enter 0 for its coefficient.
  2. Enter Divisor Coefficients: Similarly, input the coefficients of your divisor polynomial, separated by commas.
  3. Click "Calculate Division": The calculator will then display the quotient, the remainder, and a detailed breakdown of each step of the long division process.

Example:

Let's divide x^3 - 2x^2 + 5x - 1 by x - 1.

  • Dividend Coefficients: 1, -2, 5, -1 (for 1x^3 - 2x^2 + 5x^1 - 1x^0)
  • Divisor Coefficients: 1, -1 (for 1x^1 - 1x^0)

The calculator will show you the step-by-step process to arrive at the quotient and remainder.

Polynomial Long Division Calculator

Enter the coefficients of your dividend and divisor polynomials, separated by commas. For missing terms, use 0. For example, x^3 – 2x + 5 would be entered as 1, 0, -2, 5.

// Helper function to parse comma-separated coefficients function parseCoefficients(coeffStr) { var coeffs = coeffStr.split(',').map(function(s) { return parseFloat(s.trim()); }).filter(function(n) { return !isNaN(n); }); // Remove leading zeros unless it's just [0] return removeLeadingZeros(coeffs); } // Helper function to convert coefficient array to polynomial string function polyToString(coeffs, variable) { if (!coeffs || coeffs.length === 0 || (coeffs.length === 1 && coeffs[0] === 0)) { return "0"; } var degree = coeffs.length – 1; var terms = []; for (var i = 0; i 0 && terms.length > 0) { term += "+"; } else if (coeff 0) { term += variable; if (exp > 1) { term += "^" + exp; } } terms.push(term); } return terms.join(" "); } // Helper function to remove leading zeros from a coefficient array function removeLeadingZeros(coeffs) { var i = 0; while (i < coeffs.length – 1 && coeffs[i] === 0) { i++; } return coeffs.slice(i); } // Helper function to get the degree of a polynomial function getDegree(coeffs) { if (!coeffs || coeffs.length === 0 || (coeffs.length === 1 && coeffs[0] === 0)) { return -Infinity; // Degree of zero polynomial is undefined or -infinity for comparison } return coeffs.length – 1; } // Helper function to multiply a polynomial by a single term (coeff * x^exp) function polyMultiplyTerm(polyCoeffs, termCoeff, termExp) { var polyDegree = getDegree(polyCoeffs); if (polyDegree === -Infinity) { // Multiplying zero polynomial return [0]; } var resultDegree = polyDegree + termExp; var result = new Array(resultDegree + 1).fill(0); for (var i = 0; i < polyCoeffs.length; i++) { var currentPolyExp = polyDegree – i; // Exponent of current term in polyCoeffs var newExp = currentPolyExp + termExp; result[resultDegree – newExp] += polyCoeffs[i] * termCoeff; } return removeLeadingZeros(result); } // Helper function to subtract two polynomials function polySubtract(poly1Coeffs, poly2Coeffs) { var degree1 = getDegree(poly1Coeffs); var degree2 = getDegree(poly2Coeffs); if (degree1 === -Infinity && degree2 === -Infinity) return [0]; if (degree1 === -Infinity) return polyMultiplyTerm(poly2Coeffs, -1, 0); // 0 – P2 = -P2 if (degree2 === -Infinity) return poly1Coeffs; // P1 – 0 = P1 var maxDegree = Math.max(degree1, degree2); var result = new Array(maxDegree + 1).fill(0); // Pad poly1Coeffs var paddedPoly1 = new Array(maxDegree + 1).fill(0); for (var i = 0; i <= degree1; i++) { paddedPoly1[i + (maxDegree – degree1)] = poly1Coeffs[i]; } // Pad poly2Coeffs var paddedPoly2 = new Array(maxDegree + 1).fill(0); for (var i = 0; i <= degree2; i++) { paddedPoly2[i + (maxDegree – degree2)] = poly2Coeffs[i]; } for (var i = 0; i <= maxDegree; i++) { result[i] = paddedPoly1[i] – paddedPoly2[i]; } return removeLeadingZeros(result); } function calculatePolynomialDivision() { var dividendStr = document.getElementById("dividendCoeffs").value; var divisorStr = document.getElementById("divisorCoeffs").value; var resultDiv = document.getElementById("result"); resultDiv.innerHTML = ""; // Clear previous results var dividend = parseCoefficients(dividendStr); var divisor = parseCoefficients(divisorStr); if (dividend.length === 0 || divisor.length === 0) { resultDiv.innerHTML = 'Please enter valid coefficients for both polynomials.'; return; } if (divisor.length === 1 && divisor[0] === 0) { resultDiv.innerHTML = 'Division by zero polynomial is not allowed.'; return; } var dividendDegree = getDegree(dividend); var divisorDegree = getDegree(divisor); if (dividend.length === 1 && dividend[0] === 0) { // If dividend is 0 resultDiv.innerHTML = 'Quotient: ' + polyToString([0], 'x') + ''; resultDiv.innerHTML += 'Remainder: ' + polyToString([0], 'x') + ''; resultDiv.innerHTML += 'Dividing the zero polynomial results in a zero quotient and zero remainder.'; return; } if (divisorDegree > dividendDegree) { resultDiv.innerHTML = 'Quotient: ' + polyToString([0], 'x') + ''; resultDiv.innerHTML += 'Remainder: ' + polyToString(dividend, 'x') + ''; resultDiv.innerHTML += 'The degree of the divisor is greater than the degree of the dividend, so the quotient is 0 and the remainder is the dividend itself.'; return; } var currentRemainder = dividend; var quotientCoeffs = new Array(dividendDegree – divisorDegree + 1).fill(0); var stepsHtml = '

Step-by-Step Division:

'; var stepCounter = 1; while (getDegree(currentRemainder) >= divisorDegree && !(currentRemainder.length === 1 && currentRemainder[0] === 0)) { var remLeadCoeff = currentRemainder[0]; var remLeadExp = getDegree(currentRemainder); var divLeadCoeff = divisor[0]; var divLeadExp = divisorDegree; var qTermCoeff = remLeadCoeff / divLeadCoeff; var qTermExp = remLeadExp – divLeadExp; // Add this term to the quotient // quotientCoeffs index corresponds to the exponent from highest degree down // e.g., if quotient degree is 2, index 0 is x^2, index 1 is x^1, index 2 is x^0 // qTermExp = 2 -> index 0 // qTermExp = 1 -> index 1 // qTermExp = 0 -> index 2 // So, index = (quotientCoeffs.length – 1) – qTermExp quotientCoeffs[(dividendDegree – divisorDegree) – qTermExp] = qTermCoeff; // Polynomial to subtract from currentRemainder var subtractionPoly = polyMultiplyTerm(divisor, qTermCoeff, qTermExp); stepsHtml += '
'; stepsHtml += '

Step ' + stepCounter + ':

'; stepsHtml += 'Current Dividend: ' + polyToString(currentRemainder, 'x') + ''; stepsHtml += 'Divisor: ' + polyToString(divisor, 'x') + ''; stepsHtml += '1. Divide the leading term of the current dividend (' + polyToString([remLeadCoeff], 'x') + (remLeadExp > 0 ? '^' + remLeadExp : '') + ') by the leading term of the divisor (' + polyToString([divLeadCoeff], 'x') + (divLeadExp > 0 ? '^' + divLeadExp : '') + ').'; stepsHtml += ' Resulting quotient term: ' + polyToString([qTermCoeff], 'x') + (qTermExp > 0 ? '^' + qTermExp : '') + ''; stepsHtml += '2. Multiply this quotient term (' + polyToString([qTermCoeff], 'x') + (qTermExp > 0 ? '^' + qTermExp : '') + ') by the entire divisor (' + polyToString(divisor, 'x') + ').'; stepsHtml += ' Result: ' + polyToString(subtractionPoly, 'x') + ''; stepsHtml += '3. Subtract this result from the current dividend.'; stepsHtml += ' (' + polyToString(currentRemainder, 'x') + ') - (' + polyToString(subtractionPoly, 'x') + ')'; currentRemainder = polySubtract(currentRemainder, subtractionPoly); currentRemainder = removeLeadingZeros(currentRemainder); stepsHtml += ' New Remainder: ' + polyToString(currentRemainder, 'x') + ''; stepsHtml += '
'; stepCounter++; } var finalQuotient = removeLeadingZeros(quotientCoeffs); var finalRemainder = removeLeadingZeros(currentRemainder); resultDiv.innerHTML += stepsHtml; resultDiv.innerHTML += 'Final Quotient: ' + polyToString(finalQuotient, 'x') + ''; resultDiv.innerHTML += 'Final Remainder: ' + polyToString(finalRemainder, 'x') + ''; } .calculator-container { font-family: Arial, sans-serif; max-width: 700px; margin: 20px auto; padding: 20px; border: 1px solid #ccc; border-radius: 8px; background-color: #f9f9f9; } .calculator-container h2 { text-align: center; color: #333; margin-bottom: 20px; } .calculator-container p { margin-bottom: 15px; line-height: 1.5; } .calculator-input label { display: block; margin-bottom: 5px; font-weight: bold; color: #555; } .calculator-input input[type="text"] { width: calc(100% – 22px); padding: 10px; margin-bottom: 15px; border: 1px solid #ddd; border-radius: 4px; font-size: 16px; } .calculator-container button { display: block; width: 100%; padding: 12px 20px; background-color: #007bff; color: white; border: none; border-radius: 4px; font-size: 18px; cursor: pointer; transition: background-color 0.3s ease; } .calculator-container button:hover { background-color: #0056b3; } .calculator-result { margin-top: 25px; padding: 15px; border: 1px solid #e0e0e0; border-radius: 4px; background-color: #fff; min-height: 50px; } .calculator-result h3 { color: #333; margin-top: 0; margin-bottom: 10px; } .calculator-result p { margin-bottom: 8px; } .calculator-result .step { background-color: #eaf6ff; border-left: 4px solid #007bff; padding: 10px; margin-bottom: 10px; border-radius: 4px; } .calculator-result .final-answer { font-weight: bold; color: #28a745; font-size: 1.1em; margin-top: 15px; } .error-message { color: #dc3545; font-weight: bold; margin-top: 10px; } code { background-color: #eee; padding: 2px 4px; border-radius: 3px; font-family: monospace; }

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