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

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

  • $f\left(x\right)=g\left(x\right)q\left(x\right)+r\left(x\right)$

  • where

    • $q$ is the quotient

    • $r$ is the remainder

      • The degree of $r$ is less than the degree of $g$

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')

• 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 $f\left(x\right)=x^3+6x^2-9x-14$ by $g\left(x\right)=x-2$

• Start with the highest power term of the answer

  • Write out this multiplied by the divisor $g$

    • and subtract

•  Continue the process for each decreasing power term

  • multiplying by the divisor and subtracting each time

•  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 $g\left(x\right)=x-2$ is degree 1

    • So you need to continue until you have only a constant term left

• In this case the remainder is zero, so $g$ is a factor of $f$

  • So $f$ can be written in the form $f\left(x\right)=g\left(x\right)q\left(x\right)+r\left(x\right)$ as

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

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 $p\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}$

  • Use polynomial long division to carry out the division

  • That will give an answer in the form$f\left(x\right)=g\left(x\right)q\left(x\right)+r\left(x\right)$

    • so $p\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}=q\left(x\right)+\frac{r\left(x\right)}{g\left(x\right)}$

  • $g$ has a higher degree than $r$

    • so $\underset{x\to \pm \infty }{\lim }\frac{r\left(x\right)}{g\left(x\right)}=0$

  • So as $x$ increases or decreases without bound

    • $p$ gets closer and closer to the quotient $q$

  • $q\left(x\right)$ is the equation of the slant asymptote

• E.g. consider the rational function $p\left(x\right)=\frac{2x^2-7x+1}{x+1}$

  • Polynomial long division shows that $2x^2-7x+1=\left(x+1\right)\left(2x-9\right)+10$

  • Therefore

    • $p\left(x\right)=\frac{\left(x+1\right)\left(2x-9\right)+10}{\left(x+1\right)}=2x-9+\frac{10}{x+1}$

  • Considering the end behavior, $\underset{x\to \pm \infty }{\lim }\frac{10}{x+1}=0$

  • So as $x$ increases or decreases without bound

    • $p$ gets closer and closer to $2x-9$

  • $2x-9$ is the equation of the slant asymptote