Use this calculator to divide a cubic polynomial (ax³ + bx² + cx + d) by a linear factor (x – k) using synthetic division. Enter the coefficients of your dividend polynomial and the root 'k' from your divisor.
Results:
Quotient:
Remainder:
How Synthetic Division Works (Steps):
Synthetic division is a streamlined method for dividing polynomials by a linear factor of the form (x – k). It simplifies the long division process by working exclusively with the coefficients of the polynomial.
Let's illustrate the process for dividing a polynomial P(x) = ax³ + bx² + cx + d by (x – k):
Set up the division: Write down the coefficients of the dividend (a, b, c, d) in descending order of powers. To the left, place the value of 'k' from the divisor (x – k). If your divisor is (x + k), remember to use -k as your root.
Bring down the first coefficient: Bring the first coefficient (a) straight down below the line. This 'a' is the leading coefficient of your quotient.
Multiply and add (first iteration):
Multiply the 'k' value by the number you just brought down (a). Write this product (ka) directly under the second coefficient of the dividend (b).
Add the numbers in that column (b + ka). Write the sum below the line. This sum is the second coefficient of your quotient.
Repeat (second iteration):
Multiply 'k' by the new sum you just obtained (b + ka). Write this product under the third coefficient of the dividend (c).
Add the numbers in that column (c + k(b + ka)). Write the sum below the line. This sum is the third coefficient of your quotient.
Final step for remainder:
Multiply 'k' by the last sum you obtained (c + k(b + ka)). Write this product under the constant term of the dividend (d).
Add the numbers in that column (d + k(c + k(b + ka))). Write the final sum below the line. This final number is your remainder.
Form the quotient: The numbers below the line (excluding the very last one, which is the remainder) are the coefficients of your quotient polynomial. The degree of the quotient will be one less than the original dividend. For a cubic dividend (x³), the quotient will be a quadratic polynomial (ax² + bx + c).
Example: Divide (x³ – 2x + 1) by (x – 1)
Here, the dividend coefficients are a=1, b=0, c=-2, d=1. The divisor is (x – 1), so k=1.
1 | 1 0 -2 1
| 1 1 -1
-----------------
1 1 -1 0
The numbers below the line are 1, 1, -1, and 0. The first three (1, 1, -1) are the coefficients of the quotient, and the last number (0) is the remainder.
Therefore, the quotient is 1x² + 1x – 1 = x² + x – 1.
The remainder is 0.
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function calculatePolynomialDivision() {
// Get dividend coefficients
var a = parseFloat(document.getElementById('dividendCoeffA').value);
var b = parseFloat(document.getElementById('dividendCoeffB').value);
var c = parseFloat(document.getElementById('dividendCoeffC').value);
var d = parseFloat(document.getElementById('dividendCoeffD').value);
// Get divisor root
var k = parseFloat(document.getElementById('divisorRootK').value);
// Validate inputs
if (isNaN(a) || isNaN(b) || isNaN(c) || isNaN(d) || isNaN(k)) {
document.getElementById('quotientResult').innerHTML = "Please enter valid numbers for all fields.";
document.getElementById('remainderResult').innerHTML = "";
return;
}
// Perform Synthetic Division
// For dividend ax^3 + bx^2 + cx + d divided by (x – k)
// Quotient coefficients will be q1, q2, q3 for q1x^2 + q2x + q3
// Remainder will be 'remainder'
var q1 = a; // First coefficient of quotient (x^2 term)
var q2 = b + (q1 * k); // Second coefficient of quotient (x^1 term)
var q3 = c + (q2 * k); // Third coefficient of quotient (x^0 term)
var remainder = d + (q3 * k); // Final remainder
// Format quotient polynomial string
var quotientTerms = [];
if (q1 !== 0) {
var term1 = (q1 === 1 ? "" : (q1 === -1 ? "-" : q1)) + "x²";
quotientTerms.push(term1);
}
if (q2 !== 0) {
var term2 = (q2 === 1 ? "x" : (q2 === -1 ? "-x" : Math.abs(q2) + "x"));
if (q2 > 0 && quotientTerms.length > 0) {
term2 = "+" + term2;
} else if (q2 0 && quotientTerms.length > 0) {
term3 = "+" + term3;
} else if (q3 < 0) {
term3 = "-" + term3;
}
quotientTerms.push(term3);
}
var quotientString = quotientTerms.join('');
if (quotientString === "") {
quotientString = "0"; // If all quotient coefficients are zero
}
// Display results
document.getElementById('quotientResult').innerHTML = quotientString;
document.getElementById('remainderResult').innerHTML = remainder;
}
// Run calculation on page load with default values
window.onload = calculatePolynomialDivision;