Enter the coefficients of your dividend polynomial, separated by spaces. For any missing terms (e.g., no x² term in x³ + 2x – 1), use 0 as a placeholder. Then, enter the value 'k' from your divisor (x – k).
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font-family: Arial, sans-serif;
background-color: #f9f9f9;
border: 1px solid #ddd;
padding: 20px;
border-radius: 8px;
max-width: 600px;
margin: 20px auto;
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.synthetic-division-calculator h2 {
color: #333;
text-align: center;
margin-bottom: 20px;
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margin-bottom: 15px;
line-height: 1.6;
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margin-bottom: 15px;
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.calculator-input-group label {
display: block;
margin-bottom: 5px;
font-weight: bold;
color: #555;
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.calculator-input-group input[type="text"],
.calculator-input-group input[type="number"] {
width: calc(100% – 22px);
padding: 10px;
border: 1px solid #ccc;
border-radius: 4px;
font-size: 16px;
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background-color: #007bff;
color: white;
padding: 12px 20px;
border: none;
border-radius: 4px;
cursor: pointer;
font-size: 16px;
width: 100%;
box-sizing: border-box;
transition: background-color 0.3s ease;
}
.synthetic-division-calculator button:hover {
background-color: #0056b3;
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.calculator-result {
margin-top: 20px;
padding: 15px;
border: 1px solid #e0e0e0;
border-radius: 4px;
background-color: #eaf6ff;
min-height: 50px;
color: #333;
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.calculator-result p {
margin: 5px 0;
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function calculateSyntheticDivision() {
var dividendCoeffsStr = document.getElementById("dividendCoefficients").value;
var divisorKStr = document.getElementById("divisorK").value;
var resultDiv = document.getElementById("result");
resultDiv.innerHTML = ""; // Clear previous results
var dividendCoeffs = dividendCoeffsStr.split(' ').map(Number).filter(function(n) { return !isNaN(n); });
var k = Number(divisorKStr);
if (dividendCoeffs.length === 0) {
resultDiv.innerHTML = "Please enter at least one coefficient for the dividend.";
return;
}
if (isNaN(k)) {
resultDiv.innerHTML = "Please enter a valid number for the divisor value (k).";
return;
}
// Handle constant polynomial case
if (dividendCoeffs.length === 1) {
resultDiv.innerHTML = "Quotient Polynomial: 0Remainder: " + dividendCoeffs[0] + "";
return;
}
var quotientCoeffs = [];
var remainder = 0;
var currentDividend = dividendCoeffs[0];
// Bring down the first coefficient
quotientCoeffs.push(currentDividend);
for (var i = 1; i < dividendCoeffs.length; i++) {
var multipliedValue = currentDividend * k;
currentDividend = dividendCoeffs[i] + multipliedValue;
if (i < dividendCoeffs.length – 1) { // All but the last sum are quotient coefficients
quotientCoeffs.push(currentDividend);
} else { // The very last sum is the remainder
remainder = currentDividend;
}
}
// Format the quotient polynomial
var quotientPolynomial = "";
// The degree of the quotient is one less than the degree of the dividend.
// If dividend has N coefficients (degree N-1), quotient has N-1 coefficients (degree N-2).
var quotientDegree = dividendCoeffs.length – 2;
if (quotientCoeffs.length === 0) {
quotientPolynomial = "0";
} else {
var firstTermAdded = false;
for (var j = 0; j 0) {
term += " + ";
} else {
term += " – ";
}
} else if (coeff 1) {
term += "x^" + degree;
} else if (degree === 1) {
term += "x";
}
quotientPolynomial += term;
firstTermAdded = true;
}
}
if (quotientPolynomial === "") { // If all quotient coefficients were 0 (e.g., 0x^2 + 0x + 0)
quotientPolynomial = "0";
}
}
var remainderOutput = "";
if (remainder !== 0) {
remainderOutput = "Remainder: " + remainder + "";
} else {
remainderOutput = "Remainder: 0″;
}
resultDiv.innerHTML = "Quotient Polynomial: " + quotientPolynomial + "" + remainderOutput;
}
Understanding Synthetic Division
Synthetic division is a streamlined method for dividing a polynomial by a linear binomial of the form (x - k). It's a highly efficient algebraic shortcut that simplifies the process of polynomial division, making it easier to find roots, factor polynomials, and evaluate polynomial functions.
When to Use Synthetic Division
This technique is specifically applicable when your divisor is a linear factor with a leading coefficient of 1. That is, it must be in the form (x - k). If your divisor is more complex, such as (2x - 3) or (x² + 4), you would typically need to use polynomial long division or perform an initial adjustment to the divisor to fit the (x - k) format.
How Synthetic Division Works (Step-by-Step)
The process of synthetic division involves a series of multiplications and additions:
Setup: Write down only the coefficients of the dividend polynomial in descending order of powers. It's crucial to include a zero for any missing terms (e.g., if x² is absent in a cubic polynomial, use 0 as its coefficient). To the left of these coefficients, write the value of k from your divisor (x - k).
Bring Down: Bring the first coefficient of the dividend straight down below the line. This number becomes the first coefficient of your quotient.
Multiply and Add (Repeat):
Multiply the number you just brought down by k.
Write this product directly under the next coefficient of the dividend.
Add these two numbers together and write the sum below the line.
Repeat this multiply-and-add process for all remaining coefficients until you reach the end of the dividend coefficients.
Interpret the Result:
The numbers below the line (excluding the very last one) are the coefficients of your quotient polynomial, in descending order of powers. The degree of this quotient polynomial will be one less than the degree of the original dividend.
The very last number below the line is the remainder of the division.
Example Calculation
Let's illustrate by dividing the polynomial x³ - 4x² + 5x - 2 by (x - 1).
Here, the dividend coefficients are 1, -4, 5, -2, and the divisor value k = 1.
The coefficients of the quotient polynomial are 1, -3, 2. Since the original polynomial was of degree 3, the quotient polynomial will be of degree 2. Thus, the quotient is 1x² - 3x + 2, which simplifies to x² - 3x + 2.
The last number, 0, is the remainder.
Therefore, the result of (x³ - 4x² + 5x - 2) / (x - 1) is x² - 3x + 2 with a remainder of 0.
Using the Calculator
Our Synthetic Division Calculator automates this entire process. Simply input the coefficients of your dividend polynomial into the first field, ensuring you correctly represent any missing terms with zeros (e.g., for x⁴ + 3x² - 7, you would enter 1 0 3 0 -7). Then, provide the value of k from your divisor (x - k) in the second field. Click "Calculate," and the tool will instantly display the quotient polynomial and the remainder, saving you time and reducing the chance of errors in manual calculations.