12.3: Division of Polynomials
Learning Objectives
 Divide a polynomial by a monomial.
 Divide a polynomial by a binomial.
 Rewrite and graph rational functions.
Introduction
A rational expression is formed by taking the quotient of two polynomials.
Some examples of rational expressions are
Just as with rational numbers, the expression on the top is called the numerator and the expression on the bottom is called the denominator. In special cases we can simplify a rational expression by dividing the numerator by the denominator.
Divide a Polynomial by a Monomial
We’ll start by dividing a polynomial by a monomial. To do this, we divide each term of the polynomial by the monomial. When the numerator has more than one term, the monomial on the bottom of the fraction serves as the common denominator to all the terms in the numerator.
Example 1
Divide.
a)
b)
c)
Solution
a)
b)
c)
A common error is to cancel the denominator with just one term in the numerator.
Consider the quotient
Remember that the denominator of 4 is common to both the terms in the numerator. In other words we are dividing both of the terms in the numerator by the number 4.
The correct way to simplify is:
A common mistake is to cross out the number 4 from the numerator and the denominator, leaving just
Example 2
Divide
Solution
The negative sign in the denominator changes all the signs of the fractions:
Divide a Polynomial by a Binomial
We divide polynomials using a method that’s a lot like long division with numbers. We’ll explain the method by doing an example.
Example 3
Divide
Solution
When we perform division, the expression in the numerator is called the dividend and the expression in the denominator is called the divisor.
To start the division we rewrite the problem in the following form:
We start by dividing the first term in the dividend by the first term in the divisor:
We place the answer on the line above the
Next, we multiply the
Now we subtract
Now, we bring down the 5, the next term in the dividend.
And now we go through that procedure once more. First we divide the first term of
Multiply 1 by the divisor,
Subtract
Since there are no more terms from the dividend to bring down, we are done. The quotient is
Remember that for a division with a remainder the answer is
Check
To check the answer to a long division problem we use the fact that
For the problem above, here’s how we apply that fact to check our solution:
The answer checks out.
To check your answers to long division problems involving polynomials, try the solver at http://calc101.com/webMathematica/longdivide.jsp. It shows the long division steps so you can tell where you may have made a mistake.
Rewrite and Graph Rational Functions
In the last section we saw how to find vertical and horizontal asymptotes. Remember, the horizontal asymptote shows the value of
When it comes to finding asymptotes, there are basically four different types of rational functions.
Case 1: The polynomial in the numerator has a lower degree than the polynomial in the denominator.
Take, for example,
The horizontal asymptote is
Case 2: The polynomial in the numerator has the same degree as the polynomial in the denominator.
Take, for example,
So the expression can be written as
Because the denominator of the remainder is bigger than the numerator of the remainder, the remainder will approach zero for large values of
The horizontal asymptote is
Case 3: The polynomial in the numerator has a degree that is one more than the polynomial in the denominator.
Take, for example,
When the rational function approaches a straight line for large values of
Case 4: The polynomial in the numerator has a degree that in two or more than the degree in the denominator. For example:
This is actually the simplest case of all: the polynomial has no horizontal or oblique asymptotes.
Example 5
Find the horizontal or oblique asymptotes of the following rational functions.
a)
b)
c)
d)
Solution
a) When we simplify the function, we get
b) We cannot divide the two polynomials. There is a horizontal asymptote at
c) The power of the numerator is 3 more than the power of the denominator. There are no horizontal or oblique asymptotes.
d) When we simplify the function, we get
Notice that a rational function will either have a horizontal asymptote, an oblique asymptote or neither kind. In other words, a function can’t have both; in fact, it can’t have more than one of either kind. On the other hand, a rational function can have any number of vertical asymptotes at the same time that it has horizontal or oblique asymptotes.
Review Questions
Divide the following polynomials:

2x+42 
x−4x 
5x−355x 
x2+2x−5x 
4x2+12x−36−4x 
2x2+10x+72x2 
x3−x−2x2 
5x4−93x 
x3−12x2+3x−412x2 
3−6x+x3−9x3 
x2+3x+6x+1 
x2−9x+6x−1 
x2+5x+4x+4 
x2−10x+25x−5 
x2−20x+12x−3 
3x2−x+5x−2 
9x2+2x−8x+4 
3x2−43x+1 
5x2+2x−92x−1 
x2−6x−125x4
Find all asymptotes of the following rational functions:

x2x−2 
1x+4 
x2−1x2+1 
x−4x2−9 
x2+2x+14x−1 
x3+14x−1 
x−x3x2−6x−7 
x4−2x8x+24
Graph the following rational functions. Indicate all asymptotes on the graph:

x2x+2 
x3−1x2−4 
x2+12x−4 
x−x23x+2
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Feb 23, 2012Last Modified:
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