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 $f\left(x\right)$ and a non-zero polynomial divisor $d\left(x\right)$ where the degree of $d\left(x\right)$ is less than or equal to the degree of $f\left(x\right),$ there exist unique polynomials $q\left(x\right)$ and $r\left(x\right)$ such that $f\left(x\right)=d\left(x\right)q\left(x\right)+r\left(x\right)$ where $q\left(x\right)$ is the quotient and $r\left(x\right)$ is the remainder. The remainder is either equal to zero or has degree strictly less than $d\left(x\right).$

synthetic division

a shortcut method that can be used to divide a polynomial by a binomial of the form x – k