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Synthetic Division Calculator

Divide a polynomial by (x − r) using synthetic division. See the classic three-row tableau, the quotient coefficients and the remainder, step by step.

Synthetic division tableau

     3 │     1   -12     0   -42
       │           3   -27   -81
       └────────────────────────
             1    -9   -27  -123

Bring down 1; multiply by 3 and add down each column. The last number is the remainder.

Quotient & remainder

(x^3 − 12x^2 − 42) ÷ (x − 3) = x^2 − 9x − 27 + -123/(x − 3)

Remainder = -123. By the remainder theorem this equals p(3). Since it is not 0, (x − 3) is not a factor of the polynomial.

What is the Synthetic Division Calculator?

The ByteTools Synthetic Division Calculator divides any polynomial by a linear factor (x − r) using the synthetic-division shortcut: bring down the leading coefficient, multiply by r, add down the column, and repeat.

  • Classic three-row synthetic division tableau
  • Quotient coefficients and remainder written out
  • Remainder-theorem check: remainder = p(r)
  • Flags when (x − r) is an exact factor
  • Accepts decimal and negative coefficients and r values
  • 100% private — runs entirely in your browser

How to use the Synthetic Division Calculator

  1. 1

    Enter the polynomial's coefficients from highest degree to constant, using 0 for missing powers.

  2. 2

    Enter r, the value from the divisor (x − r).

  3. 3

    Read the three-row synthetic division tableau.

  4. 4

    Read the quotient polynomial and the remainder below it.

  5. 5

    Copy the full working with one click.

About the Synthetic Division Calculator

The ByteTools Synthetic Division Calculator divides any polynomial by a linear factor (x − r) using the synthetic-division shortcut: bring down the leading coefficient, multiply by r, add down the column, and repeat. The tool renders the classic three-row tableau exactly as it appears in algebra textbooks, so you can copy the working straight into homework.

Along with the tableau you get the quotient polynomial written out, the remainder, and the remainder-theorem interpretation: the remainder equals p(r), so a remainder of zero proves that (x − r) is a factor and r is a root. That makes the tool handy both for polynomial division practice and for testing candidate roots.

It runs 100% locally in your browser with JavaScript. Nothing you enter is uploaded, results update as you type, and the whole tableau can be copied as text.

Frequently asked questions

How does synthetic division work?

Write the coefficients in a row and r to the left. Bring the first coefficient down, multiply it by r and write the product under the next coefficient, add the column, and repeat to the end. The bottom row holds the quotient coefficients, and the final number is the remainder.

When can you use synthetic division instead of long division?

Synthetic division only works when dividing by a linear factor of the form (x − r). For divisors like x² + 1 or 2x − 3 in unsimplified form you need polynomial long division. To divide by (2x − 3), you can use r = 3/2 and then divide the quotient's coefficients by 2.

What does the remainder tell you?

By the remainder theorem, dividing p(x) by (x − r) leaves remainder p(r). So the remainder is the polynomial's value at r. If the remainder is 0, then r is a root and (x − r) is a factor — which is why synthetic division is the standard way to test candidate rational roots.

How do I handle missing terms like x³ + 8?

Insert a zero for every missing power. x³ + 8 becomes coefficients 1, 0, 0, 8. Skipping the zeros misaligns the columns and produces a wrong quotient, so the tool asks for the complete coefficient list and shows a preview of the polynomial it parsed.

Can r be negative or a fraction?

Yes. Dividing by (x + 2) means r = −2, since x + 2 = x − (−2). Fractions like 0.5 work too. The tableau shows the exact multiplications and additions performed with your value of r.

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