Synthetic Division Calculator (table, quotient & remainder for x − k)

Understanding the synthetic division calculator — concept and principle

This calculator divides a polynomial by a linear factor x − k (or ax − b) with synthetic division. It shows the three rows of the table, the quotient, the remainder, the result written out and whether the divisor is a factor, with a remainder-theorem check. Use it for algebra homework, factoring, finding roots and evaluating polynomials quickly.

How it is calculated

1. Write k (for x − 3, k = 3; for x + 2, k = −2) and the coefficients of P(x) from the highest power down, with 0 for missing powers.
2. Bring the first coefficient down.
3. Multiply it by k, write the product under the next coefficient and add.
4. Repeat to the end.

The bottom row gives the quotient's coefficients, one degree lower, and the last number is the remainder. By the remainder theorem, the remainder equals P(k). For ax − b, the table uses k = b/a and the quotient is then divided by a.

The principle

Synthetic division is long division by x − k with everything unnecessary removed. In long division by x − k, every step multiplies the divisor by a single term and subtracts; the x's are just placeholders, and subtracting −k is the same as adding k. Keeping only the coefficients and adding instead of subtracting gives the short table. It is also a fast way to evaluate P(k): the steps are the same as Horner's method, which rewrites the polynomial as nested multiplications.

Worked example

Default inputs: P(x) = 2x³ − 7x² + 4x + 5, divisor x − 3, so k = 3.

1. Row 1 (coefficients): 2, −7, 4, 5.
2. Bring down 2. 2 × 3 = 6; −7 + 6 = −1.
3. −1 × 3 = −3; 4 + (−3) = 1. 1 × 3 = 3; 5 + 3 = 8.
4. Row 3: 2, −1, 1 │ 8. Quotient 2x² − x + 1, remainder 8.
5. Check: P(3) = 54 − 63 + 12 + 5 = 8. The remainder is not 0, so x − 3 is not a factor: P(x) = (x − 3)(2x² − x + 1) + 8.

How to read the result

A remainder of 0 means k is a root and x − k is a factor; you can then continue with the quotient to find more roots. A non-zero remainder is the value P(k), which is useful for graphing or checking. The quotient always has degree one less than P.

Common mistakes

• Using −3 instead of 3 for the divisor x − 3.
• Leaving out zero coefficients for missing powers.
• Subtracting instead of adding in each column.
• Using synthetic division for a divisor of degree 2 or more.

Frequently asked questions

When can I use synthetic division?

When dividing by a linear expression of the form x − k (or ax − b, with an extra division by a).

What does a zero remainder mean?

k is a root of the polynomial, and x − k is one of its factors.

How is synthetic division related to Horner's method?

They perform the same arithmetic; Horner's method uses it to evaluate P(k).

Sources: OpenStax, College Algebra 2e (2021), §5.4 Dividing Polynomials and §5.5 Zeros of Polynomial Functions