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Division Algorithm

from class:

Algebra and Trigonometry

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 $g(x)$.

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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.
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