Skip to main content

Precalculus for Team-Based Inquiry Learning: PREVIEW Edition — Instructor Version

TBIL Community

Section 4.4 Polynomial Long Division (PR4)

Subsection 4.4.1 Activities

Observation 4.4.1.

We have seen previously that we can reduce rational functions by factoring, for example
\begin{equation*} \dfrac{x^2+5x+4}{x^3+3x^2+2x}=\dfrac{(x+1)(x+4)}{x(x+2)(x+1)}=\dfrac{x+4}{x(x+2)}. \end{equation*}
In this section, we will explore the question: what can we do to simplify rational functions if we are not able to reduce by easily factoring?

Definition 4.4.2.

Recall that a fraction is called proper if its numerator is smaller than its denominator, and improper if the numerator is larger than the denominator (so \(\dfrac{3}{5}\) is a proper fraction, but \(\dfrac{32}{7}\) is an improper fraction). Similarly, we define a proper rational function to be a rational function where the degree of the numerator is less than the degree of the denominator.

Activity 4.4.3.

Label each of the following rational functions as either proper or improper.
  1. \(\displaystyle \dfrac{x^3+x}{x^2+4}\)
  2. \(\displaystyle \dfrac{3}{x^2+3x+4}\)
  3. \(\displaystyle \dfrac{7+x^3}{x^2+x+1}\)
  4. \(\displaystyle \dfrac{x^4+x+1}{x^4+4x^2}\)
Answer.
A, C, and D are improper, while B is proper.

Observation 4.4.4.

When dealing with an improper fraction such as \(\dfrac{32}{7}\text{,}\) it is sometimes useful to rewrite this as an integer plus a proper fraction, e.g. \(\dfrac{32}{7}=4+\dfrac{4}{7}\text{.}\) Similarly, it will sometimes be useful to rewrite an improper rational function as the sum of a polynomial and a proper rational function, such as \(\dfrac{x^3+x}{x^2+4}=x-\dfrac{3x}{x^2+4}\text{.}\)

Activity 4.4.5.

Consider the improper fraction \(\dfrac{357}{11}\text{.}\)
(a)
Use long division to write \(\dfrac{357}{11}\) as an integer plus a proper fraction.
Answer.
So \(\dfrac{357}{11}=32+\dfrac{5}{11}\text{.}\)
(b)
Now we will carefully redo this process in a way that we can generalize to rational functions. Note that we can rewrite \(357\) as \(357=3\cdot10^2+5\cdot10+7\text{,}\) and \(11\) as \(11=1\cdot10+1\text{.}\) By comparing the leading terms in these expansions, we see that to knock off the leading term of \(357\text{,}\) we need to multiply \(11\) by \(3\cdot10^1\text{.}\)
Using the fact that \(357=11\cdot30+27\text{,}\) rewrite \(\dfrac{357}{11}\) as \(\dfrac{357}{11}=30+\dfrac{?}{11}\text{.}\)
Answer.
\(\dfrac{357}{11}=30+\dfrac{27}{11}\text{.}\)
(c)
Note now that if we can rewrite \(\dfrac{27}{11}\) as an integer plus a proper fraction, we will be done, since \(\dfrac{357}{11}=30+\dfrac{27}{11}\text{.}\)
Rewrite \(\dfrac{27}{11}=?+\dfrac{?}{11}\) as an integer plus a proper fraction.
Answer.
\(\dfrac{27}{11}=2+\dfrac{5}{11}\text{.}\)
(d)
Combine your work in the previous two parts to rewrite \(\dfrac{357}{11}\) as an integer plus a proper fraction. How does this compare to what you obtained in part (a)?
Answer.
\(\dfrac{357}{11}=30+\dfrac{27}{11}=30+2+\dfrac{5}{11}=32+\dfrac{5}{11}\text{.}\)

Activity 4.4.6.

Now let’s consider the rational function \(\dfrac{3x^2+5x+7}{x+1}\text{.}\) We want to rewrite this as a polynomial plus a proper rational function.
(a)
Looking at the leading terms, what do we need to multiply \(x+1\) by so that it would have the same leading term as \(3x^2+5x+7\text{?}\)
  1. \(\displaystyle 3\)
  2. \(\displaystyle x\)
  3. \(\displaystyle 3x\)
  4. \(\displaystyle 3x+5\)
Answer.
C
(b)
Rewrite \(3x^2+5x+7=3x(x+1)+?\text{,}\) and use this to rewrite \(\dfrac{3x^2+5x+7}{x+1}=3x+\dfrac{?}{x+1}\text{.}\)
Answer.
\(\dfrac{3x^2+5x+7}{x+1}=3x+\dfrac{2x+7}{x+1}\)
(c)
Now focusing on \(\dfrac{2x+7}{x+1}\text{,}\) what do we need to multiply \(x+1\) by so that it would have the same leading term as \(2x+7\text{?}\)
  1. \(\displaystyle 2\)
  2. \(\displaystyle x\)
  3. \(\displaystyle 2x\)
  4. \(\displaystyle 2x+7\)
Answer.
A
(d)
Rewrite \(\dfrac{2x+7}{x+1}=2+\dfrac{?}{x+1}\text{.}\)
Answer.
\(\dfrac{2x+7}{x+1}=2+\dfrac{5}{x+1}\)
(e)
Combine this with the previous parts to rewrite \(\dfrac{3x^2+5x+7}{x+1}=3x+?+\dfrac{?}{x+1}\text{.}\)
Answer.
\(\dfrac{3x^2+5x+7}{x+1}=3x+2+\dfrac{5}{x+1}\)

Activity 4.4.7.

Next we will use the notation of long division to rewrite the rational function \(\dfrac{3x^2+5x+7}{x+1}\) as a polynomial plus a proper rational function.
(a)
First, let’s use long division notation to write the quotient.
What do we need to multiply \(x+1\) by so that it would have the same leading term as \(3x^2+5x+7\text{?}\)
Answer.
(b)
Now to rewrite \(3x^2+5x+7\) as \(3x(x+1)+?\text{,}\) place the product \(3x(x+1)\) below and subtract.
Answer.
(c)
Now focusing on \(2x+7\text{,}\) what do we need to multiply \(x+1\) by so that it would have the same leading term as \(2x+7\text{?}\)
Answer.
(d)
Now, subtract \(2(x+1)\) to finish the long division.
Answer.
(e)
This long division calculation has shown that
\begin{equation*} 3x^2+5x+7 = (x+1)(3x+2)+5\text{.} \end{equation*}
Use this to rewrite \(\dfrac{3x^2+5x+7}{x+1}\) as a polynomial plus a proper rational function.
Answer.
\(\dfrac{3x^2+5x+7}{x+1} = 3x+2+\dfrac{5}{x+1}\)

Activity 4.4.9.

Rewrite \(\dfrac{x^2+1}{x-1}\) as a polynomial plus a proper rational function.
Hint.
Note that \(x^2+1=x^2+0x+1\text{.}\)
Answer.
\(x+1+\dfrac{3}{x-1}\text{.}\)

Activity 4.4.10.

Rewrite \(\dfrac{x^5+x^3+2x^2-6x+7}{x^2+x-1}\) as a polynomial plus a proper rational function.
Answer.
\(x^3-x^2+3x-2+\dfrac{-x+5}{x^2+x-1}\text{.}\)

Activity 4.4.11.

Rewrite \(\dfrac{3x^4-5x^2+2}{x-1}\) as a polynomial plus a proper rational function.
Answer.
\(3x^3+3x^2-2x-2\text{.}\)

Exercises 4.4.2 Exercises

PrevTopNext