Polynomial Division

Exam code: 4037

Written by: Ashika|Reviewed by: Caroline Carroll|Updated 2 July 2026

Polynomial Division

Polynomial division

Polynomial division is the process of dividing two polynomials
- This is usually only useful when the degree of the denominator is less than or equal to the degree of the numerator

Polynomial division is a method for splitting polynomials into factor pairs
- (with or without a remainder term)

The main uses of polynomial division are
- factorising polynomials
- simplifying 'top-heavy' algebraic fractions

The method used for polynomial division is just like the long division method for numbers
- sometimes called the 'bus stop method'

The answer to a polynomial division question is built up term by term
- Start by dividing by the highest power term
- Write out this multiplied by the divisor and subtract

Continue to divide by each reducing power term and subtracting your answer each time

If the divisor is not a factor of the polynomial then there will be a remainder term left at the end of the division

Worked example

For the polynomial $f (x) = x^{4} + 11 x^{2} - 1$ divide $f (x)$ by $x + 3$ and write the remainder.

Set up the polynomial division ('bus stop')

There is no $x^{3}$ term so write this as $0 x^{3}$ in the method. There is no $x$ term so write this as $0 x$ in the method.

The first division step to consider is $x^{4} \div x$ .

.Multiply $x^{3}$ by $x + 3$ and subtract from $x^{4} + 0 x$ .

Bring the $11 x^{2}$ down and divide $- 3 x^{3}$ by $x$ . Continue with each step until you are finished.

The remainder is 179.

Polynomial division usually involves dividing by a linear term
- a term of the form $(x + p)$ where $p$ is a constant and usually an integer
It is possible to divide a polynomial by a quadratic term (and cubic, etc)
- this would be a term of the form $(x^{2} + q x + r)$ where $q$ and $r$ are constants
- this is what is meant by a quadratic divisor

The process is the same as for a linear divisor
- However, as $x^{2}$ will not divide into $x$ (in the polynomial division sense at least) the remainder, if there is one, could be of linear form, i.e. $(r x + s)$ where $r$ and $s$ are constants
  - It is possible that $r = 0$ and so the remainder is still a constant