Synthetic Division Calculator

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Synthetic Division Calculator
Divide by (x – k)Divide by (x + k)

(separated by commas, e.g., 2, -3, 0, 5)
Result:
Enter coefficients and k to see the result.
function calculateSynthetic(){var coeffStr=document.getElementById('coeffs').value;var kValStr=document.getElementById('k_val').value;var divType=document.getElementById('div_type').value;var showSteps=document.getElementById('showSteps').checked;if(!coeffStr||!kValStr){alert('Please enter coefficients and a value for k.');return;}var coeffs=coeffStr.split(',').map(function(item){return parseFloat(item.trim());});var k=parseFloat(kValStr);if(coeffs.some(isNaN)||isNaN(k)){alert('Please enter valid numeric values.');return;}if(divType==='plus'){k=-k;}var resultRow=[];var multiplicationRow=[0];var n=coeffs.length;resultRow[0]=coeffs[0];for(var i=1;i<n;i++){multiplicationRow[i]=resultRow[i-1]*k;resultRow[i]=coeffs[i]+multiplicationRow[i];}var quotient=resultRow.slice(0,-1);var remainder=resultRow[n-1];var html='
';html+='Quotient Coefficients: '+quotient.join(', ')+'
';html+='Remainder: '+remainder+'
';var polyStr=";for(var j=0;j0 && j>0)?'+':";var term=c;if(c===1 && power>0)term=";if(c===-1 && power>0)term='-';if(power===0){polyStr+=' '+sign+c;}else if(power===1){polyStr+=' '+sign+term+'x';}else{polyStr+=' '+sign+term+'x'+power+'';}}if(!polyStr)polyStr='0′;html+='
Quotient: '+polyStr;if(remainder!==0){html+=' + ('+remainder+' / '+(divType==='minus'?'x – ':'x + ')+Math.abs(parseFloat(kValStr))+')';}html+='
';if(showSteps){html+='
';for(var a=0;a<coeffs.length;a++){html+='';}html+='';for(var b=0;b<multiplicationRow.length;b++){html+='';}html+='';for(var c=0;c<resultRow.length;c++){html+='';}html+='
'+k+''+coeffs[a]+'
'+(b===0?":multiplicationRow[b])+'
'+resultRow[c]+'
';}document.getElementById('answer').innerHTML=html;}

How to Use the Synthetic Division Calculator

The synthetic division calculator is a specialized tool designed to simplify the process of dividing a polynomial by a linear factor of the form (x – k) or (x + k). Unlike standard long division, which can be cumbersome and prone to errors, synthetic division provides a streamlined, tabular approach for finding the quotient and remainder.

To use this calculator, follow these simple steps:

Divisor Format
Select whether you are dividing by (x – k) or (x + k). This ensures the correct sign for the root $k$ is used in the calculation.
Polynomial Coefficients
Enter the coefficients of your dividend polynomial in descending order of power. Use commas to separate them. Crucial: If a power of x is missing (e.g., x³ + 5, where x² and x¹ are missing), you must enter 0 as a placeholder for those terms.
Value of k
Enter the constant value $k$ from your linear divisor.

The Synthetic Division Method

Synthetic division is a shorthand method of polynomial division. It only works when dividing by a linear factor. The math behind the calculator follows this logic:

P(x) / (x – k) = Q(x) + R / (x – k)

  • Step 1: Write down the coefficients of the polynomial and the value of $k$ (the zero of the divisor).
  • Step 2: Drop the first coefficient down to the bottom row.
  • Step 3: Multiply the bottom row value by $k$ and place the result in the next column's middle row.
  • Step 4: Add the top row coefficient and the middle row value, placing the sum in the bottom row.
  • Step 5: Repeat until all columns are filled. The final value in the bottom row is the remainder.

Calculation Example

Example: Divide 2x³ – 6x² + 0x – 8 by (x – 3).

Step-by-step solution:

  1. Identify k: Since divisor is (x – 3), k = 3.
  2. Identify Coefficients: 2, -6, 0, -8.
  3. Bring down 2: Bottom row starts with 2.
  4. Multiply 2 * 3 = 6. Add to next coefficient: -6 + 6 = 0.
  5. Multiply 0 * 3 = 0. Add to next coefficient: 0 + 0 = 0.
  6. Multiply 0 * 3 = 0. Add to final coefficient: -8 + 0 = -8.
  7. Result: Quotient is 2x² + 0x + 0, Remainder is -8.

Common Questions

Can I use synthetic division for divisors like x² + 1?

No. Synthetic division is specifically designed for linear divisors (degree 1). For quadratic divisors or higher, you must use polynomial long division.

What if the divisor is (2x – 4)?

You can still use synthetic division, but you must first normalize the divisor. Divide both the dividend and divisor by the lead coefficient of the divisor (in this case, 2), turning the divisor into (x – 2). Perform the division, but remember the final quotient coefficients may need adjustment depending on the context.

Why is the remainder important?

According to the Remainder Theorem, the remainder of P(x) / (x – k) is equal to P(k). This makes the synthetic division calculator an excellent tool for evaluating functions quickly.

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