# Polynomial and Rational Functions

## Dividing Polynomials


### Learning Objectives

1. Dividing polynomials using long division (IA 5.4.3)
2. Dividing polynomials using synthetic division (IA 5.4.4)

### Objective 1: Dividing polynomials using long division (IA 5.4.3)

To divide a polynomial by a binomial, we follow a procedure very similar to long division of numbers. So, let’s look carefully at the steps we take when we divide a 3-digit number, 875, by a 2-digit number, 25.


### Practice Makes Perfect

Vocabulary of the example. 
Fill in the blanks.

### Practice Makes Perfect

Sometimes division of polynomials, just like division of numbers, leaves a remainder. We write the remainder as a fraction with the divisor as the denominator.

Also, if you look back at the dividends in previous examples, you will notice that the terms were written in descending order of degrees, and there were no missing degrees.


### Practice Makes Perfect


Dividing polynomials using long division.


### Objective 2: Dividing polynomials using synthetic division (IA 5.4.4)

As you probably noticed, long division can be tedious. Synthetic division uses the patterns from long division as a basis to make a process much simpler by leaving the variable terms out. The same example in synthetic division format is shown next.

Synthetic division only works when the divisor is of the form (x−c).

### Practice Makes Perfect


Dividing polynomials using synthetic division.

The exterior of the Lincoln Memorial in Washington, D.C., is a large rectangular solid with length 61.5 meters (m), width 40 m, and height 30 m.National Park Service. "Lincoln Memorial Building Statistics." http://www.nps.gov/linc/historyculture/lincoln-memorial-building-statistics.htm. Accessed 4/3/2014 We can easily find the volume using elementary geometry.
So the volume is 73,800 cubic meters 
   
   Suppose we knew the volume, length, and width. We could divide to find the height.
As we can confirm from the dimensions above, the height is 30 m. We can use similar methods to find any of the missing dimensions. We can also use the same method if any, or all, of the measurements contain variable expressions. For example, suppose the volume of a rectangular solid is given by the polynomial 
   
   The length of the solid is given by 
 
   the width is given by 
        To find the height of the solid, we can use polynomial division, which is the focus of this section.

### Using Long Division to Divide Polynomials

We are familiar with the long division algorithm for ordinary arithmetic. We begin by dividing into the digits of the dividend that have the greatest place value. We divide, multiply, subtract, include the digit in the next place value position, and repeat. For example, let’s divide 178 by 3 using long division.
Another way to look at the solution is as a sum of parts. This should look familiar, since it is the same method used to check division in elementary arithmetic.
We call this the Division Algorithm and will discuss it more formally after looking at an example.
Division of polynomials that contain more than one term has similarities to long division of whole numbers. We can write a polynomial dividend as the product of the divisor and the quotient added to the remainder. The terms of the polynomial division correspond to the digits (and place values) of the whole number division. This method allows us to divide two polynomials. For example, if we were to divide 
         by 
          using the long division algorithm, it would look like this:
We have found
or
We can identify the dividend, the divisor, the quotient, and the remainder.
Writing the result in this manner illustrates the Division Algorithm.

### Using Synthetic Division to Divide Polynomials

As we’ve seen, long division of polynomials can involve many steps and be quite cumbersome. Synthetic division is a shorthand method of dividing polynomials for the special case of dividing by a linear factor whose leading coefficient is 1.
To illustrate the process, recall the example at the beginning of the section.
Divide 
 
by 
 
using the long division algorithm.
The final form of the process looked like this:
There is a lot of repetition in the table. If we don’t write the variables but, instead, line up their coefficients in columns under the division sign and also eliminate the partial products, we already have a simpler version of the entire problem.
Synthetic division carries this simplification even a few more steps. Collapse the table by moving each of the rows up to fill any vacant spots. Also, instead of dividing by 2, as we would in division of whole numbers, then multiplying and subtracting the middle product, we change the sign of the “divisor” to –2, multiply and add. The process starts by bringing down the leading coefficient.
We then multiply it by the “divisor” and add, repeating this process column by column, until there are no entries left. The bottom row represents the coefficients of the quotient; the last entry of the bottom row is the remainder. In this case, the quotient is 
  and the remainder is 
 
The process will be made more clear in .

### Using Polynomial Division to Solve Application Problems

Polynomial division can be used to solve a variety of application problems involving expressions for area and volume. We looked at an application at the beginning of this section. Now we will solve that problem in the following example.

### Key Equations


### Key Concepts

1. Polynomial long division can be used to divide a polynomial by any polynomial with equal or lower degree. See  and .
2. The Division Algorithm tells us that a polynomial dividend can be written as the product of the divisor and the quotient added to the remainder.
3. Synthetic division is a shortcut that can be used to divide a polynomial by a binomial in the form 
 
See , , and .
4. Polynomial division can be used to solve application problems, including area and volume. See .

### Section Exercises


### Verbal


### Algebraic 

For the following exercises, use long division to divide. Specify the quotient and the remainder.
For the following exercises, use synthetic division to find the quotient. Ensure the equation is in the form required by synthetic division. (Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor.)
For the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization.

### Graphical

For the following exercises, use the graph of the third-degree polynomial and one factor to write the factored form of the polynomial suggested by the graph. The leading coefficient is one.
For the following exercises, use synthetic division to find the quotient and remainder.

### Technology

For the following exercises, use a calculator with CAS to answer the questions.

### Extensions

For the following exercises, use synthetic division to determine the quotient involving a complex number.

### Real-World Applications

For the following exercises, use the given length and area of a rectangle to express the width algebraically.
For the following exercises, use the given volume of a box and its length and width to express the height of the box algebraically.
For the following exercises, use the given volume and radius of a cylinder to express the height of the cylinder algebraically.