Synthetic division

Synthetic division is a compact procedure for dividing a polynomial by a linear factor of the form x − c. It uses the factor’s root, c, to produce the quotient’s coefficients and the remainder.

Polynomial long division: A division algorithm that repeatedly subtracts multiples of a divisor polynomial to find a quotient and remainder. It performs the same division with explicit polynomial terms rather than a compressed coefficient table.
Polynomial coefficient: A numerical or symbolic multiplier of a power of the variable in a polynomial. Synthetic division operates on coefficients rather than writing each polynomial term.
Polynomial factorization: The expression of a polynomial as a product of simpler polynomial factors. A zero remainder tests whether a proposed linear factor belongs in a factorization.
Euclidean division of polynomials: The division principle expressing one polynomial as a divisor times a quotient plus a lower-degree remainder. Synthetic division is a specialized, compressed case of this general division principle.
Remainder theorem: The theorem that dividing a polynomial by x − c leaves remainder f(c). The final entry in synthetic division is f(c), the remainder predicted by this theorem.
Polynomial degree: The greatest exponent with a nonzero coefficient in a polynomial. The degree determines how many quotient coefficients the procedure produces.
Rational root theorem: A theorem restricting the possible rational roots of a polynomial with integer coefficients. Its candidate roots can be tested efficiently by synthetic division.
Non-monic linear divisor: A linear polynomial whose leading coefficient is not 1. The standard coefficient shortcut assumes a monic divisor and needs adjustment otherwise.
Factor theorem: The theorem that x − c divides a polynomial f(x) exactly when f(c) = 0. A zero remainder in the procedure certifies that the divisor is a factor.
Polynomial root: A value that makes a polynomial equal zero when substituted for its variable. The procedure uses the root c corresponding to the divisor x − c.

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