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Study Guides > College Algebra CoRequisite Course

Summary: Dividing Polynomials

Key Equations

Division Algorithm [latex]f\left(x\right)=d\left(x\right)q\left(x\right)+r\left(x\right)[/latex] where [latex]q\left(x\right)\ne 0[/latex]

Key Concepts

  • Polynomial long division can be used to divide a polynomial by any polynomial with equal or lower degree.
  • The Division Algorithm tells us that a polynomial dividend can be written as the product of the divisor and the quotient added to the remainder.
  • Synthetic division is a shortcut that can be used to divide a polynomial by a binomial of the form x – k.
  • Polynomial division can be used to solve application problems, including area and volume.


Division Algorithm
given a polynomial dividend [latex]f\left(x\right)[/latex] and a non-zero polynomial divisor [latex]d\left(x\right)[/latex] where the degree of [latex]d\left(x\right)[/latex] is less than or equal to the degree of [latex]f\left(x\right)[/latex], there exist unique polynomials [latex]q\left(x\right)[/latex] and [latex]r\left(x\right)[/latex] such that [latex]f\left(x\right)=d\left(x\right)q\left(x\right)+r\left(x\right)[/latex] where [latex]q\left(x\right)[/latex] is the quotient and [latex]r\left(x\right)[/latex] is the remainder. The remainder is either equal to zero or has degree strictly less than [latex]d\left(x\right)[/latex].
synthetic division
a shortcut method that can be used to divide a polynomial by a binomial of the form x k

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  • College Algebra. Provided by: OpenStax Authored by: Abramson, Jay et al.. Located at: https://openstax.org/books/college-algebra/pages/1-introduction-to-prerequisites. License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/[email protected].