Polynomial division is a fundamental algebraic process used to divide one polynomial by another. It's analogous to long division for numbers and is essential for solving various problems in algebra, calculus, and engineering. The goal is to find a quotient polynomial and a remainder polynomial, such that:
Dividend = Divisor × Quotient + Remainder
The degree of the remainder must be less than the degree of the divisor. When the remainder is zero, it means the divisor is a factor of the dividend.
Why Use Polynomial Division?
Factoring Polynomials: If you can find a root of a polynomial (e.g., using the Rational Root Theorem), you can use polynomial division to divide the polynomial by (x - root). This reduces the degree of the polynomial, making it easier to find the remaining roots.
Simplifying Rational Expressions: Polynomial division can simplify complex fractions involving polynomials by separating a whole part from a proper fraction.
Finding Asymptotes: In calculus, the quotient of polynomial division can reveal slant (oblique) asymptotes for rational functions.
Solving Equations: It's a key step in solving higher-degree polynomial equations.
How it Works: The Long Division Method
The process of polynomial long division involves a series of steps: division, multiplication, and subtraction, repeated until the remainder's degree is less than the divisor's degree.
Set up: Write the dividend and divisor in standard form (descending powers of the variable), leaving spaces or placeholders (e.g., 0xn) for missing terms.
Divide the leading terms: Divide the leading term of the dividend by the leading term of the divisor. This is the first term of the quotient.
Multiply: Multiply the first term of the quotient by the entire divisor.
Subtract: Subtract the result from the dividend. Bring down the next term of the dividend.
Repeat: Use the new polynomial (the result of the subtraction) as the new dividend and repeat steps 2-4 until the degree of the resulting polynomial is less than the degree of the divisor.
Result: The final result is expressed as Quotient + Remainder / Divisor.
Example:
Let's divide 3x3 - 5x2 + 0x - 1 by x - 2:
Step 1: Divide 3x3 by x to get 3x2 (first term of quotient).
Step 2: Multiply 3x2 by (x - 2) to get 3x3 - 6x2.
Step 3: Subtract (3x3 - 6x2) from (3x3 - 5x2) to get x2. Bring down the 0x. New dividend is x2 + 0x.
Step 4: Divide x2 by x to get x (second term of quotient).
Step 5: Multiply x by (x - 2) to get x2 - 2x.
Step 6: Subtract (x2 - 2x) from (x2 + 0x) to get 2x. Bring down the -1. New dividend is 2x - 1.
Step 7: Divide 2x by x to get 2 (third term of quotient).
Step 8: Multiply 2 by (x - 2) to get 2x - 4.
Step 9: Subtract (2x - 4) from (2x - 1) to get 3.
The quotient is 3x2 + x + 2 and the remainder is 3. So the result is 3x2 + x + 2 + 3/(x - 2).
function performPolynomialDivision() {
var dividendStr = document.getElementById("dividend").value.replace(/\s+/g, ");
var divisorStr = document.getElementById("divisor").value.replace(/\s+/g, ");
var resultDiv = document.getElementById("result");
var stepsDiv = document.getElementById("steps");
resultDiv.innerHTML = "";
stepsDiv.innerHTML = "
Steps:
";
if (!dividendStr || !divisorStr) {
resultDiv.innerHTML = "Please enter both dividend and divisor polynomials.";
return;
}
try {
var parsedDividend = parsePolynomial(dividendStr);
var parsedDivisor = parsePolynomial(divisorStr);
if (parsedDivisor.degree = 0 ? " + (" + formatPolynomial(remainder) + ")/(" + formatPolynomial(parsedDivisor) + ")" : "");
if (resultString === "") resultString = "0"; // Handle cases where quotient and remainder are zero
resultDiv.innerHTML = resultString;
} catch (e) {
resultDiv.innerHTML = "Error parsing polynomials: " + e.message;
}
}
function parsePolynomial(polyStr) {
var coefficients = {};
var currentPolyStr = polyStr;
var regex = /([+-]?\d*)?x(?:\^(\d+))?|([+-]?\d+)/g;
var match;
var maxDegree = -1;
// Handle cases like 'x', '-x', '5', '-3' without explicit powers
if (!polyStr.includes('x')) {
var constantMatch = polyStr.match(/^([+-]?\d+)$/);
if (constantMatch) {
return { degree: 0, coeffs: { 0: parseInt(constantMatch[1]) } };
} else if (polyStr === "") {
return { degree: -1, coeffs: {} }; // Represent zero polynomial
}
}
while ((match = regex.exec(polyStr)) !== null) {
var coeffStr = match[1];
var exp = match[2] ? parseInt(match[2]) : (match[0].includes('x') ? 1 : 0);
var constant = match[3];
var coeff;
if (constant) { // It's a constant term
coeff = parseInt(constant);
exp = 0;
} else {
if (coeffStr === "" || coeffStr === "+") coeff = 1;
else if (coeffStr === "-") coeff = -1;
else coeff = parseInt(coeffStr);
}
if (isNaN(coeff)) {
throw new Error("Invalid coefficient in '" + polyStr + "'");
}
if (isNaN(exp)) {
throw new Error("Invalid exponent in '" + polyStr + "'");
}
coefficients[exp] = (coefficients[exp] || 0) + coeff;
if (exp > maxDegree) maxDegree = exp;
}
// Ensure all degrees up to maxDegree exist, even if 0
for (var i = 0; i <= maxDegree; i++) {
if (!(i in coefficients)) {
coefficients[i] = 0;
}
}
// Find the actual highest degree with a non-zero coefficient
var actualMaxDegree = -1;
for (var deg in coefficients) {
if (coefficients[deg] !== 0) {
actualMaxDegree = Math.max(actualMaxDegree, parseInt(deg));
}
}
return { degree: actualMaxDegree, coeffs: coefficients };
}
function formatPolynomial(poly) {
if (!poly || Object.keys(poly.coeffs).length === 0 || poly.degree b – a);
for (var i = 0; i 0 ? "+" : "-";
var absCoeff = Math.abs(coeff);
var term = "";
if (degree === 0) {
term = absCoeff.toString();
} else if (degree === 1) {
term = (absCoeff === 1 ? "" : absCoeff.toString()) + "x";
} else {
term = (absCoeff === 1 ? "" : absCoeff.toString()) + "x^" + degree;
}
if (terms.length === 0) { // First term
term = (coeff < 0 ? "-" : "") + term;
} else {
term = (sign === "+" ? " + " : " – ") + term;
}
terms.push(term);
}
return terms.join("");
}
function longDivision(dividend, divisor) {
var steps = [];
var quotient = { degree: -1, coeffs: {} };
var remainder = JSON.parse(JSON.stringify(dividend)); // Deep copy
steps.push("Dividend: " + formatPolynomial(dividend));
steps.push("Divisor: " + formatPolynomial(divisor));
steps.push("—");
if (divisor.degree < 0) { // Divisor is zero polynomial
throw new Error("Divisor cannot be zero.");
}
if (dividend.degree = divisor.degree && remainder.degree >= 0) {
var leadingTermDividendCoeff = remainder.coeffs[remainder.degree] || 0;
var leadingTermDivisorCoeff = divisor.coeffs[divisor.degree] || 0;
if (leadingTermDivisorCoeff === 0) { // Should not happen if divisor is parsed correctly
throw new Error("Divisor leading coefficient is zero.");
}
var quotientCoeff = leadingTermDividendCoeff / leadingTermDivisorCoeff;
var quotientDegree = remainder.degree – divisor.degree;
// Add to quotient
quotient.coeffs[quotientDegree] = (quotient.coeffs[quotientDegree] || 0) + quotientCoeff;
quotient.degree = Math.max(quotient.degree, quotientDegree);
// Subtract (quotientTerm * divisor) from remainder
var termToSubtract = { degree: -1, coeffs: {} };
for (var deg in divisor.coeffs) {
var d = parseInt(deg);
var c = divisor.coeffs[deg];
var newDegree = d + quotientDegree;
var newCoeff = c * quotientCoeff;
termToSubtract.coeffs[newDegree] = (termToSubtract.coeffs[newDegree] || 0) + newCoeff;
termToSubtract.degree = Math.max(termToSubtract.degree, newDegree);
}
steps.push("Current Remainder: " + formatPolynomial(remainder));
steps.push("Leading Term of Current Remainder: " + (remainder.coeffs[remainder.degree] > 0 ? "+" : "") + remainder.coeffs[remainder.degree] + "x^" + remainder.degree);
steps.push("Leading Term of Divisor: " + (divisor.coeffs[divisor.degree] > 0 ? "+" : "") + divisor.coeffs[divisor.degree] + "x^" + divisor.degree);
steps.push("Quotient Term: " + (quotientCoeff > 0 ? "+" : "") + quotientCoeff + "x^" + quotientDegree);
steps.push("Multiplying (" + (quotientCoeff > 0 ? "+" : "") + quotientCoeff + "x^" + quotientDegree + ") by Divisor (" + formatPolynomial(divisor) + "): " + formatPolynomial(termToSubtract));
steps.push("Subtracting: (" + formatPolynomial(remainder) + ") – (" + formatPolynomial(termToSubtract) + ")");
var newRemainder = { degree: -1, coeffs: {} };
var allDegrees = new Set([…Object.keys(remainder.coeffs).map(Number), …Object.keys(termToSubtract.coeffs).map(Number)]);
for (var deg of allDegrees) {
var remCoeff = remainder.coeffs[deg] || 0;
var subCoeff = termToSubtract.coeffs[deg] || 0;
var diff = remCoeff – subCoeff;
if (diff !== 0) {
newRemainder.coeffs[deg] = diff;
newRemainder.degree = Math.max(newRemainder.degree, deg);
}
}
remainder = newRemainder;
if (remainder.degree < 0) remainder.degree = -1; // Ensure correct handling of zero remainder
steps.push("Resulting Remainder: " + formatPolynomial(remainder));
steps.push("—");
// Re-calculate max degree for remainder for next iteration
var actualMaxDegree = -1;
for (var deg in remainder.coeffs) {
if (remainder.coeffs[deg] !== 0) {
actualMaxDegree = Math.max(actualMaxDegree, parseInt(deg));
}
}
remainder.degree = actualMaxDegree;
}
// Ensure quotient is properly formatted (no zero coefficients)
var finalQuotient = { degree: -1, coeffs: {} };
for(var deg in quotient.coeffs) {
if (quotient.coeffs[deg] !== 0) {
finalQuotient.coeffs[deg] = quotient.coeffs[deg];
finalQuotient.degree = Math.max(finalQuotient.degree, parseInt(deg));
}
}
return { quotient: finalQuotient, remainder: remainder, steps: steps };
}