Objectives
- Rewrite a rational function as a polynomial plus a proper rational function.
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*}
\frac{x^2+5x+4}{x^3+3x^2+2x}=\frac{(x+1)(x+4)}{x(x+2)(x+1)}=\frac{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 \(\frac{3}{5}\) is a proper fraction, but \(\frac{32}{7}\) 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.- \(\displaystyle \frac{x^3+x}{x^2+4}\)
- \(\displaystyle \frac{3}{x^2+3x+4}\)
- \(\displaystyle \frac{7+x^3}{x^2+x+1}\)
- \(\displaystyle \frac{x^4+x+1}{x^4+4x^2}\)
Answer.
A, C, and D are improper, while B is proper.
Observation 4.4.4.
When dealing with an improper fraction such as \(\frac{32}{7}\text{,}\) it is sometimes useful to rewrite this as an integer plus a proper fraction, e.g. \(\frac{32}{7}=4+\frac{4}{7}\text{.}\) 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 \(\frac{x^3+x}{x^2+4}=x-\frac{3x}{x^2+4}\text{.}\)
Activity 4.4.5.
Consider the improper fraction \(\frac{357}{11}\text{.}\)
(a)
Use long division to write \(\frac{357}{11}\) as an integer plus a proper fraction.
Answer.
So \(\frac{357}{11}=32+\frac{5}{11}\text{.}\)
(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 \(\frac{357}{11}\) as \(\frac{357}{11}=30+\frac{?}{11}\text{.}\)
Answer.
\(\frac{357}{11}=30+\frac{27}{11}\text{.}\)
(c)
Note now that if we can rewrite \(\frac{27}{11}\) as an integer plus a proper fraction, we will be done, since \(\frac{357}{11}=30+\frac{27}{11}\text{.}\)
Rewrite \(\frac{27}{11}=?+\frac{?}{11}\) as an integer plus a proper fraction.
Answer.
\(\frac{27}{11}=2+\frac{5}{11}\text{.}\)
(d)
Combine your work in the previous two parts to rewrite \(\frac{357}{11}\) as an integer plus a proper fraction. How does this compare to what you obtained in part (a)?
Answer.
\(\frac{357}{11}=30+\frac{27}{11}=30+2+\frac{5}{11}=32+\frac{5}{11}\text{.}\)
Activity 4.4.6.
Now let’s consider the rational function \(\frac{3x^2+5x+7}{x+1}\text{.}\) 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 \(3x^2+5x+7\text{?}\)
- \(\displaystyle 3\)
- \(\displaystyle x\)
- \(\displaystyle 3x\)
- \(\displaystyle 3x+5\)
Answer.
C
(b)
Rewrite \(3x^2+5x+7=3x(x+1)+?\text{,}\) and use this to rewrite \(\frac{3x^2+5x+7}{x+1}=3x+\frac{?}{x+1}\text{.}\)
Answer.
\(\frac{3x^2+5x+7}{x+1}=3x+\frac{2x+7}{x+1}\)
(c)
Now focusing on \(\frac{2x+7}{x+1}\text{,}\) what do we need to multiply \(x+1\) by so that it would have the same leading term as \(2x+7\text{?}\)
- \(\displaystyle 2\)
- \(\displaystyle x\)
- \(\displaystyle 2x\)
- \(\displaystyle 2x+7\)
Answer.
A
(d)
Rewrite \(\frac{2x+7}{x+1}=2+\frac{?}{x+1}\text{.}\)
Answer.
\(\frac{2x+7}{x+1}=2+\frac{5}{x+1}\)
(e)
Combine this with the previous parts to rewrite \(\frac{3x^2+5x+7}{x+1}=3x+?+\frac{?}{x+1}\text{.}\)
Answer.
\(\frac{3x^2+5x+7}{x+1}=3x+2+\frac{5}{x+1}\)
Activity 4.4.7.
Next we will use the notation of long division to rewrite the rational function \(\frac{3x^2+5x+7}{x+1}\) 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 \(\frac{3x^2+5x+7}{x+1}\) as a polynomial plus a proper rational function.
Answer.
\(\frac{3x^2+5x+7}{x+1} = 3x+2+\frac{5}{x+1}\)
Remark 4.4.8.
Note that in Activity 4.4.6 and Activity 4.4.7 we performed the same computations, but just organized our work a little differently.
Activity 4.4.9.
Rewrite \(\frac{x^2+1}{x-1}\) as a polynomial plus a proper rational function.
Hint.
Note that \(x^2+1=x^2+0x+1\text{.}\)
Answer.
\(x+1+\frac{3}{x-1}\text{.}\)
Activity 4.4.10.
Rewrite \(\frac{x^5+x^3+2x^2-6x+7}{x^2+x-1}\) as a polynomial plus a proper rational function.
Answer.
\(x^3-x^2+3x-2+\frac{-x+5}{x^2+x-1}\text{.}\)
Activity 4.4.11.
Rewrite \(\frac{3x^4-5x^2+2}{x-1}\) as a polynomial plus a proper rational function.
Answer.
\(3x^3+3x^2-2x-2\text{.}\)
Exercises 4.4.2 Exercises
Exercises available at
https://teambasedinquirylearning.github.io/precalculus/2024/exercises/#/bank/PR4/.Subsection 4.4.3 Videos
It would be great to include videos down here, like in the Calculus book!
