Use this calculator to perform synthetic division on a polynomial by a linear divisor of the form (x – k).
function calculateSyntheticDivision() {
var coefficientsInput = document.getElementById("coefficients").value;
var kValueInput = document.getElementById("kValue").value;
var resultDiv = document.getElementById("syntheticDivisionResult");
// Clear previous results
resultDiv.innerHTML = "";
// Validate kValue
var k = parseFloat(kValueInput);
if (isNaN(k)) {
resultDiv.innerHTML = "Please enter a valid number for the divisor value (k).";
return;
}
// Parse coefficients
var coeffsStrArray = coefficientsInput.split(',').map(function(item) {
return item.trim();
});
var coefficients = [];
for (var i = 0; i < coeffsStrArray.length; i++) {
var num = parseFloat(coeffsStrArray[i]);
if (isNaN(num)) {
resultDiv.innerHTML = "Please enter valid comma-separated numbers for polynomial coefficients.";
return;
}
coefficients.push(num);
}
if (coefficients.length === 0) {
resultDiv.innerHTML = "Please enter at least one coefficient.";
return;
}
// Perform Synthetic Division
var quotientCoefficients = [];
var remainder = 0;
var currentCoeffs = coefficients.slice(); // Make a copy to modify
// Bring down the first coefficient
quotientCoefficients.push(currentCoeffs[0]);
for (var j = 1; j < currentCoeffs.length; j++) {
var product = quotientCoefficients[j – 1] * k;
var sum = currentCoeffs[j] + product;
quotientCoefficients.push(sum);
}
// The last element of quotientCoefficients is the remainder
remainder = quotientCoefficients.pop();
// Format the quotient polynomial
var quotientString = "";
var degree = quotientCoefficients.length – 1;
var firstTermAdded = false;
if (quotientCoefficients.length === 0) {
quotientString = "0";
} else {
for (var l = 0; l 0) {
term += " + ";
} else {
term += " – ";
coeff = Math.abs(coeff);
}
} else { // This is the first non-zero term
if (coeff 1) {
term += "x^" + currentTermDegree;
}
quotientString += term;
}
}
// If after loop, no terms were added (e.g., all coefficients were 0)
if (!firstTermAdded) {
quotientString = "0";
}
resultDiv.innerHTML = "
Synthetic division is a simplified method for dividing a polynomial by a linear binomial of the form (x - k). It's a powerful shortcut compared to long division, especially useful for finding roots of polynomials, factoring polynomials, and evaluating polynomial functions (using the Remainder Theorem).
When to Use Synthetic Division
You can use synthetic division specifically when your divisor is a linear factor, meaning it can be written as (x - k). For example, (x - 3), (x + 2) (which is x - (-2)), or even (2x - 4) if you first factor out the 2 to get 2(x - 2) and then divide by (x - 2) and adjust the quotient later.
How Synthetic Division Works (The Algorithm)
Set up the problem: Write down the value of k (from x - k) to the left. To the right, write down only the coefficients of the dividend polynomial, making sure to include zeros for any missing terms (e.g., if x^3 + 5x - 2, coefficients are 1, 0, 5, -2).
Bring down the first coefficient: Drop the first coefficient below the line.
Multiply and Add: Multiply the number you just brought down by k, and write the result under the next coefficient. Add these two numbers together and write the sum below the line.
Repeat: Continue this process of multiplying the latest sum by k and adding it to the next coefficient until all coefficients have been processed.
Interpret the results: The last number below the line is the remainder. The other numbers below the line are the coefficients of the quotient polynomial, which will have a degree one less than the original dividend polynomial.
Using the Synthetic Division Calculator
Our calculator simplifies this process for you:
Enter Polynomial Coefficients: In the "Polynomial Coefficients" field, enter the coefficients of your dividend polynomial, separated by commas. Ensure you include a zero for any missing terms. For example, for 3x^4 - 2x^2 + 5x - 1, you would enter 3, 0, -2, 5, -1.
Enter Divisor Value (k): In the "Divisor Value (k)" field, enter the value of k from your linear divisor (x - k). If your divisor is (x + 3), then k = -3. If it's (x - 5), then k = 5.
Calculate: Click the "Calculate Synthetic Division" button.
View Results: The calculator will display the quotient polynomial and the remainder.
Example Calculation
Let's divide the polynomial x^3 - 6x^2 + 11x - 6 by (x - 1).