Summary: Divide Polynomials

Key Equations

Division Algorithm

f\left(x\right)=d\left(x\right)q\left(x\right)+r\left(x\right)

where

q\left(x\right)\ne 0

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.

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