Dividing Polynomials Step by Step Calculator

Polynomial Division Calculator

Dividend Polynomial (P(x)):

Use 'x' for the variable, '^' for exponents. Example: 3x^2 - 5x + 1

Divisor Polynomial (D(x)):

Example: x + 1 or 2x^2 - 1

// Helper to normalize polynomial representation (remove zero coeffs, sort by exp) function normalizePoly(poly) { var newPoly = {}; var exponents = Object.keys(poly).map(Number).sort(function(a, b) { return b – a; }); // Sort descending for (var i = 0; i < exponents.length; i++) { var exp = exponents[i]; var coeff = poly[exp]; if (coeff !== 0) { newPoly[exp] = coeff; } } return newPoly; } // Helper to parse a polynomial string into an object {exponent: coefficient} function parsePolynomial(polyString) { var poly = {}; if (!polyString || polyString.trim() === '') { return poly; // Return empty poly for empty string } // Normalize string: remove spaces, ensure leading + for positive terms, handle implicit 1s polyString = polyString.replace(/\s+/g, ''); if (polyString.charAt(0) !== '-' && polyString.charAt(0) !== '+') { polyString = '+' + polyString; } // This regex captures terms like +3x^2, -x, +5, -x^3 var terms = polyString.match(/([+-]?[0-9.]*x(\^[0-9]+)?|[+-]?[0-9.]+)/g); if (!terms) { return poly; // No valid terms found } for (var i = 0; i < terms.length; i++) { var term = terms[i]; var coeff = 1; var exp = 0; // Check for 'x' if (term.includes('x')) { // Extract coefficient part (e.g., "+3", "-") var coeffPart = term.split('x')[0]; if (coeffPart === '+' || coeffPart === '') { coeff = 1; } else if (coeffPart === '-') { coeff = -1; } else { coeff = parseFloat(coeffPart); } // Extract exponent part (e.g., "^2", "") var expPartMatch = term.match(/\^([0-9]+)/); if (expPartMatch) { exp = parseInt(expPartMatch[1]); } else { exp = 1; // x without exponent means x^1 } } else { // Constant term coeff = parseFloat(term); exp = 0; } // Add to polynomial, combining like terms poly[exp] = (poly[exp] || 0) + coeff; } return normalizePoly(poly); } // Helper to format a polynomial object back to a string function formatPolynomial(poly) { var terms = []; var exponents = Object.keys(poly).map(Number).sort(function(a, b) { return b – a; }); // Sort descending if (exponents.length === 0 || (exponents.length === 1 && poly[exponents[0]] === 0)) { return "0"; // Represents the zero polynomial } for (var i = 0; i 0 && terms.length > 0) { termStr += "+"; } else if (coeff = divisorLeading.exp && getLeadingTerm(remainder).coeff !== 0) { var remainderLeading = getLeadingTerm(remainder); // Calculate term to add to quotient var termCoeff = remainderLeading.coeff / divisorLeading.coeff; var termExp = remainderLeading.exp – divisorLeading.exp; var currentQuotientTerm = { coeff: termCoeff, exp: termExp }; // Add term to quotient quotient = polyAdd(quotient, { [termExp]: termCoeff }); // Multiply current quotient term by divisor var product = polyMultiplyTerm(divisor, currentQuotientTerm); // Subtract product from remainder remainder = polySubtract(remainder, product); } return { quotient: normalizePoly(quotient), remainder: normalizePoly(remainder) }; } function calculatePolynomialDivision() { var dividendString = document.getElementById("dividendPoly").value; var divisorString = document.getElementById("divisorPoly").value; var resultDiv = document.getElementById("polynomialResult"); // Input validation if (!dividendString || dividendString.trim() === "") { resultDiv.innerHTML = "Please enter the Dividend Polynomial."; return; } if (!divisorString || divisorString.trim() === "") { resultDiv.innerHTML = "Please enter the Divisor Polynomial."; return; } var dividendPoly = parsePolynomial(dividendString); var divisorPoly = parsePolynomial(divisorString); // Check for zero divisor var divisorLeading = getLeadingTerm(divisorPoly); if (divisorLeading.coeff === 0 && divisorLeading.exp === -Infinity) { // This means divisor is "0" or empty resultDiv.innerHTML = "Divisor cannot be zero."; return; } var divisionResult = dividePolynomials(dividendPoly, divisorPoly); if (divisionResult.error) { resultDiv.innerHTML = "" + divisionResult.error + ""; return; } var formattedQuotient = formatPolynomial(divisionResult.quotient); var formattedRemainder = formatPolynomial(divisionResult.remainder); var formattedDividend = formatPolynomial(dividendPoly); var formattedDivisor = formatPolynomial(divisorPoly); var outputHTML = "

Division Result:

"; outputHTML += "Quotient (Q(x)): " + formattedQuotient + ""; outputHTML += "Remainder (R(x)): " + formattedRemainder + ""; outputHTML += "This means: " + formattedDividend + " = (" + formattedQuotient + ")(" + formattedDivisor + ") + (" + formattedRemainder + ")"; resultDiv.innerHTML = outputHTML; } .calculator-container { background-color: #f9f9f9; border: 1px solid #ddd; padding: 20px; border-radius: 8px; max-width: 600px; margin: 20px auto; font-family: Arial, sans-serif; } .calculator-container h2 { text-align: center; color: #333; margin-bottom: 20px; } .calculator-input { margin-bottom: 15px; } .calculator-input label { display: block; margin-bottom: 5px; font-weight: bold; color: #555; } .calculator-input input[type="text"] { width: calc(100% – 22px); padding: 10px; border: 1px solid #ccc; border-radius: 4px; font-size: 16px; } .input-hint { font-size: 0.9em; color: #777; margin-top: 5px; } .calculate-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; } .calculate-button:hover { background-color: #0056b3; } .calculator-result { margin-top: 20px; padding: 15px; border: 1px solid #e0e0e0; border-radius: 4px; background-color: #eaf4ff; color: #333; } .calculator-result h3 { color: #007bff; margin-top: 0; } .calculator-result p { margin-bottom: 8px; line-height: 1.5; } .calculator-result code { background-color: #f0f0f0; padding: 2px 4px; border-radius: 3px; font-family: 'Courier New', Courier, monospace; }

Understanding Polynomial Division: A Step-by-Step Guide

Polynomial division is a fundamental operation in algebra, similar to long division with numbers. It allows us to divide one polynomial (the dividend) by another polynomial (the divisor) to find a quotient and a remainder. This process is crucial for factoring polynomials, finding roots, simplifying rational expressions, and understanding the behavior of polynomial functions.

What is a Polynomial?

Before diving into division, let's quickly define a polynomial. A polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. For example, 3x^4 - 2x^2 + 5x - 7 is a polynomial.

The Division Algorithm for Polynomials

Just like with integers, when you divide a polynomial P(x) by a non-zero polynomial D(x), you get a unique quotient Q(x) and a remainder R(x) 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.

Why is Polynomial Division Important?

Factoring Polynomials: If the remainder is zero, it means the divisor is a factor of the dividend. This is key for finding the roots of polynomial equations.
Finding Roots: Once a factor is found, you can reduce the degree of the polynomial, making it easier to find its roots.
Simplifying Rational Expressions: Polynomial division can simplify complex fractions involving polynomials.
Graphing Polynomials: Understanding factors and roots helps in sketching the graph of a polynomial function.

How to Perform Polynomial Long Division (Step-by-Step Process)

The process of polynomial long division mirrors numerical long division. Let's walk through an example to illustrate the steps. Suppose we want to divide P(x) = x^3 - 2x^2 - 5x + 6 by D(x) = x - 3.

Set up the Division: Write the dividend and divisor in the long division format. Ensure both polynomials are written in descending powers of the variable. If any power is missing, include it with a coefficient of zero (e.g., x^2 + 1 becomes x^2 + 0x + 1).
_______
x - 3 | x^3 - 2x^2 - 5x + 6
Divide the Leading Terms: Divide the first term of the dividend (x^3) by the first term of the divisor (x). Write the result (x^2) as the first term of the quotient.
x^2____
x - 3 | x^3 - 2x^2 - 5x + 6
Multiply the Quotient Term by the Divisor: Multiply the term you just wrote in the quotient (x^2) by the entire divisor (x - 3). Write the result (x^3 - 3x^2) below the dividend, aligning like terms.
x^2____
x - 3 | x^3 - 2x^2 - 5x + 6
-(x^3 - 3x^2)
Subtract: Subtract the product from the dividend. Remember to change the signs of all terms being subtracted. Bring down the next term from the dividend.
x^2____
x - 3 | x^3 - 2x^2 - 5x + 6
-(x^3 - 3x^2)
_________
x^2 - 5x
Repeat the Process: Now, treat the new polynomial (x^2 - 5x) as your new dividend. Divide its leading term (x^2) by the leading term of the divisor (x). Write the result (+x) in the quotient.
x^2 + x__
x - 3 | x^3 - 2x^2 - 5x + 6
-(x^3 - 3x^2)
_________
x^2 - 5x
-(x^2 - 3x)
Subtract Again:
x^2 + x__
x - 3 | x^3 - 2x^2 - 5x + 6
-(x^3 - 3x^2)
_________
x^2 - 5x
-(x^2 - 3x)
_________
-2x + 6
Repeat Until Remainder Degree is Less than Divisor Degree: Divide -2x by x, which is -2. Add -2 to the quotient. Multiply -2 by (x - 3) to get -2x + 6. Subtract this from -2x + 6.
x^2 + x - 2
x - 3 | x^3 - 2x^2 - 5x + 6
-(x^3 - 3x^2)
_________
x^2 - 5x
-(x^2 - 3x)
_________
-2x + 6
-(-2x + 6)
_________
0

In this example, the quotient Q(x) = x^2 + x - 2 and the remainder R(x) = 0. Since the remainder is 0, (x - 3) is a factor of x^3 - 2x^2 - 5x + 6.

Using the Polynomial Division Calculator

Our calculator simplifies this process for you. Simply enter your dividend polynomial and your divisor polynomial into the respective fields. The calculator will instantly provide you with the quotient and the remainder, following the same mathematical principles outlined above.

Input Tips:

Use 'x' as your variable.
Use '^' for exponents (e.g., x^3).
Coefficients of 1 can be omitted (e.g., x^2 instead of 1x^2).
Constant terms are treated as x^0 (e.g., 5).
Ensure correct signs for terms (e.g., -2x^2).

This tool is perfect for checking your homework, verifying complex calculations, or simply exploring polynomial relationships.