Division Algorithm

Definition

The Division Algorithm states that for any polynomials $f(x)$ and $g(x) \neq 0$, there exist unique polynomials $q(x)$ (quotient) and $r(x)$ (remainder) such that $f(x) = g(x)q(x) + r(x)$, where the degree of $r(x)$ is less than the degree of $g(x)$.

5 Must Know Facts For Your Next Test

1. The Division Algorithm is applicable to all polynomials as long as the divisor polynomial is non-zero.
2. The remainder, $r(x)$, must have a degree less than the degree of the divisor polynomial, $g(x).
3. The quotient and remainder obtained are unique for given polynomials.
4. If the remainder, $r(x)$, is zero, then $g(x)$ is a factor of $f(x).
5. The Division Algorithm can be used to perform polynomial long division and synthetic division.

Review Questions

• What condition must be met regarding the degrees of the remainder and divisor in the Division Algorithm?
• If you perform polynomial division using the Division Algorithm and get a zero remainder, what does it indicate about your divisor?
• Explain why or why not you can use the Division Algorithm if your divisor polynomial is zero.

Related terms

Polynomial Long Division: A method for dividing a polynomial by another polynomial of lower or equal degree, resulting in a quotient and a remainder.

Synthetic Division: A simplified form of polynomial division used specifically when dividing by a linear factor.

Remainder Theorem:

States that if a polynomial $f(x)$ is divided by $(x - c),$ then the remainder is $f(c).