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 in the form x – k.
• Polynomial division can be used to solve application problems, including area and volume.

Glossary

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