Polynomials: Division, Remainder Theorem, and Related Concepts (Vocabulary)

Vocabulary flashcards covering key concepts from polynomial division, the remainder and factor theorems, and related ideas.

Polynomial

An expression in one variable composed of terms with coefficients and powers of the variable, typically written as an x^n + … + a1 x + a_0.

Long Division (polynomials)

A method for dividing polynomials by another polynomial to obtain a quotient and a remainder, with deg(R) < deg(divisor).

Synthetic Division

A shortcut division method used when dividing by (x − r); it uses the root r to quickly compute the quotient and remainder.

Divisor

The polynomial by which another polynomial is divided (e.g., x − r).

Dividend

The polynomial that is being divided by the divisor.

Quotient

The polynomial obtained from division, not including the remainder.

Remainder

The leftover value after division; its degree is less than the degree of the divisor.

Remainder Theorem

If P(x) is divided by (x − r), the remainder is P(r).

Factor Theorem

If P(r) = 0, then (x − r) is a factor of P(x) and P(x) is divisible by (x − r).

Division Algorithm for Polynomials

P(x) = (divisor)·Q(x) + R with deg(R) < deg(divisor).

Standard Form of a Polynomial

Expression arranged in descending powers of x with numerical coefficients.

Degree of a Polynomial

The highest exponent of x with a nonzero coefficient.

Leading Coefficient

The coefficient of the highest-degree term in a polynomial.

Evaluate a Polynomial

Compute P(a) by substituting x = a into P(x).

Example Division (P(x) by x − 3)

Dividing P(x) = 2x^3 − 9x^2 + 13x − 12 by (x − 3) yields Q(x) = 2x^2 − 3x + 4 and R = 0.

Root

A value r such that P(r) = 0; corresponds to x − r being a factor of P(x).