Synthetic Division Calculator

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Perform fast polynomial division by a linear factor (x - c).

Division Input

Polynomial Coefficients (comma separated, high to low)

Divisor Constant (c in x - c)

Division Results

-2

Remainder

Calculated using the synthetic division algorithm.

Quotient Coefficients

1, -2, -4

Polynomial Result

x² - 2x - 4 + -2/(x-2)

About this calculator

What is Synthetic Division?

Synthetic division is a shorthand method of polynomial division, even simpler than long division, and usually used only for dividing a polynomial by a linear factor of the form (x - c). It uses only the coefficients, making it faster and less error-prone.

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Pro Tips

The divisor must be in the form (x - c). If you have (x + 2), c is -2.
You must include a zero coefficient for any missing terms in the polynomial (e.g., x³ + 1 has coefficients 1, 0, 0, 1).
The last number in the results row is always the remainder.

Fun Facts

"Synthetic division can be used to quickly evaluate a polynomial at x = c (Remainder Theorem)."
"It was popularized in the 19th century as a tool for finding roots of equations."
"If the remainder is zero, then (x - c) is a factor of the polynomial."

Formula: P(x) = (x - c)Q(x) + R