AP®MathsCollege BoardPrecalculusRevision NotesPolynomial & Rational FunctionsRational FunctionsPolynomial Long Division

Polynomial Long Division (College Board AP® Precalculus): Revision Note

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 9 April 2026

Polynomial long division

What is polynomial long division?

Polynomial long division is a method for splitting polynomials into factor pairs (with or without an accompanying remainder term)
- You can use it to factor polynomials
- to help simplify algebraic fractions
- or to find the equation of the slant asymptote for a rational function
If a polynomial $f$ is divided by a polynomial $g$ then $f$ can be rewritten as
- $f (x) = g (x) q (x) + r (x)$
- where
  - $q$ is the quotient
  - $r$ is the remainder
    - The degree of $r$ is less than the degree of $g$

Math equations showing polynomial factorisation. The top equation is factored fully, the bottom has a factor pair with a remainder of 10.

How do I perform polynomial long division?

The method used for polynomial long division is just like the method used to divide regular numbers
- i.e. the long division method (sometimes called 'bus stop division')

Long division of 4836 by 39 equals 124, showing step-by-step subtraction: 39, 78, 156 with multiplication tags 1×39, 2×39, 4×39.

The answer to a polynomial long division question is built up term by term
- Working downwards in powers of the variable (usually x)
E.g. to divide $f (x) = x^{3} + 6 x^{2} - 9 x - 14$ by $g (x) = x - 2$

Start with the highest power term of the answer
- Write out this multiplied by the divisor $g$
  - and subtract

Maths diagram showing division of polynomial f(x) = x^3 + 6x^2 - 9x - 14 by x - 2 with explanation boxes detailing steps.

Continue the process for each decreasing power term
- multiplying by the divisor and subtracting each time

Polynomial long division showing x^3 + 6x^2 - 9x - 14 divided by x - 2, with steps and commentary on dealing with terms.

Continue until what you are left with has a lower degree than what you are dividing by
- Then what you are left with will be the remainder $r$
  - If the divisor is a factor of the polynomial, the remainder will be zero
- In this case $g (x) = x - 2$ is degree 1
  - So you need to continue until you have only a constant term left

Polynomial long division showing x^3 + 6x^2 - 9x - 14 divided by x - 2, resulting in x^2 + 8x + 7 with remainder zero.

In this case the remainder is zero, so $g$ is a factor of $f$
- So $f$ can be written in the form $f (x) = g (x) q (x) + r (x)$ as

$\begin{array}{rcl} x^{3} + 6 x^{2} - 9 x - 14 & = & (x - 2) (x^{2} + 8 x + 7) + 0 \\ = & (x - 2) (x^{2} + 8 x + 7) \end{array}$

Examiner Tips and Tricks

Don't rush when doing algebraic division.

Finding and fixing a mistake can take longer than taking the time to do it right the first time!

How can I use polynomial long division to find the equation of the slant asymptote of a rational function?

For a rational function, if the degree of the numerator exceeds the degree of the denominator by exactly 1
- then the graph of the rational function has a slant asymptote
- See the End Behavior of Rational Functions study guide
To find the equation of the slant asymptote for a rational function $p (x) = \frac{f (x)}{g (x)}$
- Use polynomial long division to carry out the division
- That will give an answer in the form $f (x) = g (x) q (x) + r (x)$
  - so $p (x) = \frac{f (x)}{g (x)} = q (x) + \frac{r (x)}{g (x)}$
- $g$ has a higher degree than $r$
  - so $\lim_{x \to \pm \infty} \frac{r (x)}{g (x)} = 0$
- So as $x$ increases or decreases without bound
  - $p$ gets closer and closer to the quotient $q$
- $q (x)$ is the equation of the slant asymptote
E.g. consider the rational function $p (x) = \frac{2 x^{2} - 7 x + 1}{x + 1}$
- Polynomial long division shows that $2 x^{2} - 7 x + 1 = (x + 1) (2 x - 9) + 10$
- Therefore
  - $p (x) = \frac{(x + 1) (2 x - 9) + 10}{(x + 1)} = 2 x - 9 + \frac{10}{x + 1}$
- Considering the end behavior, $\lim_{x \to \pm \infty} \frac{10}{x + 1} = 0$
- So as $x$ increases or decreases without bound
  - $p$ gets closer and closer to $2 x - 9$
- $2 x - 9$ is the equation of the slant asymptote

Worked Example

The function $f$ is given by $f (x) = \frac{3 x^{3} - 4 x^{2} - 15 x + 7}{x^{2} - 5}$ .

Find the equation of the slant asymptote on the graph of $f$ .

Answer:

Carry out polynomial long division to find the quotient and remainder when the numerator is divided by the denominator

Polynomial long division solving (3x^3 - 4x^2 - 15x + 7) divided by (x^2 - 5) showing steps. Quotient is 3x-4 and final remainder is -13.

Therefore

$3 x^{3} - 4 x^{2} - 15 x + 7 = (x^{2} - 5) (3 x - 4) - 13$

and

$f (x) = \frac{(x^{2} - 5) (3 x - 4) - 13}{x^{2} - 5} = 3 x - 4 - \frac{13}{x^{2} - 5}$

Consider the end behavior

$\lim_{x \to \pm \infty} \frac{13}{x^{2} - 5} = 0$

So as $x$ increases or decreases without limit, $f (x)$ gets closer and closer to $3 x - 4$

The slant asymptote has equation $y = 3 x - 4$

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