Polynomial Division Calculator

Synthetic division is a shorthand algorithm for dividing a polynomial by a linear factor of the form (x - r). It avoids writing out the full long division process and works entirely with the coefficients.

The algorithm: write down the coefficients in order from highest to lowest degree, including zeros for any missing terms. Bring down the leading coefficient. Multiply it by r, add it to the next coefficient, write the result, repeat. The final number is the remainder; everything before it is the quotient’s coefficients.

For example, dividing x^3 - 3x^2 + 0x + 2 by (x - 2): coefficients are [1, -3, 0, 2], r = 2. Bring down 1. Multiply by 2 to get 2, add to -3 to get -1. Multiply by 2 to get -2, add to 0 to get -2. Multiply by 2 to get -4, add to 2 to get -2. So the quotient is x^2 - x - 2 with remainder -2.

If the remainder is 0, then r is a root of the polynomial, and (x - r) divides it evenly. This is the Factor Theorem: p(r) = 0 if and only if (x - r) is a factor of p(x).

Synthetic division only works for linear divisors. For quadratic divisors like (x^2 + 1), you need polynomial long division instead.

One practical use: after finding one root of a cubic or quartic by guessing or by the rational zeros theorem, synthetic division reduces the polynomial’s degree by one. A cubic with one known root becomes a quadratic, which you can solve with the quadratic formula.

The Remainder Theorem is a quick consequence: the remainder from dividing p(x) by (x - r) is exactly p(r). So synthetic division doubles as a fast way to evaluate a polynomial at a specific point, especially useful when r is not an integer.