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3.5 Dividing polynomials

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In this section, you will:
  • Use long division to divide polynomials.
  • Use synthetic division to divide polynomials.
Lincoln Memorial.
Lincoln Memorial, Washington, D.C. (credit: Ron Cogswell, Flickr)

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.

V = l ⋅ w ⋅ h    = 61.5 ⋅ 40 ⋅ 30    = 73,800

So the volume is 73,800 cubic meters   ( m ³ ) .   Suppose we knew the volume, length, and width. We could divide to find the height.

h = V l ⋅ w    = 73 , 800 61.5 ⋅ 40    = 30

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   3 x 4 − 3 x 3 − 33 x 2 + 54 x .   The length of the solid is given by   3 x ;   the width is given by   x − 2.   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.

Steps of long division for intergers.

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.

dividend =  ( divisor  ⋅  quotient )  + remainder                             178 = ( 3 ⋅ 59 ) + 1                                               = 177 + 1                                               = 178

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   2 x 3 − 3 x 2 + 4 x + 5   by   x + 2   using the long division algorithm, it would look like this: Steps of long division for polynomials.

We have found

2 x 3 − 3 x 2 + 4 x + 5 x + 2 = 2 x 2 − 7 x + 18 − 31 x + 2

or

2 x 3 − 3 x 2 + 4 x + 5 = ( x + 2 ) ( 2 x 2 − 7 x + 18 ) − 31

We can identify the dividend , the divisor , the quotient , and the remainder .

Identifying the dividend, divisor, quotient and remainder of the polynomial 2x^3-3x^2+4x+5, which is the dividend.

Writing the result in this manner illustrates the Division Algorithm.

The division algorithm

The Division Algorithm    states that, given a polynomial dividend   f ( x )   and a non-zero polynomial divisor   d ( x )   where the degree of   d ( x )   is less than or equal to the degree of   f ( x ) ,   there exist unique polynomials   q ( x )   and   r ( x )   such that

f ( x ) = d ( x ) q ( x ) + r ( x )

q ( x )   is the quotient and   r ( x )   is the remainder. The remainder is either equal to zero or has degree strictly less than   d ( x ) .  

If   r ( x ) = 0 ,   then   d ( x )   divides evenly into   f ( x ) .   This means that, in this case, both   d ( x )   and   q ( x )   are factors of   f ( x ) .  

Questions & Answers

for the "hiking" mix, there are 1,000 pieces in the mix, containing 390.8 g of fat, and 165 g of protein. if there is the same amount of almonds as cashews, how many of each item is in the trail mix?
ADNAN Reply
linear speed of an object
Melissa Reply
an object is traveling around a circle with a radius of 13 meters .if in 20 seconds a central angle of 1/7 Radian is swept out what are the linear and angular speed of the object
Melissa
test
Matrix
how to find domain
Mohamed Reply
like this: (2)/(2-x) the aim is to see what will not be compatible with this rational expression. If x= 0 then the fraction is undefined since we cannot divide by zero. Therefore, the domain consist of all real numbers except 2.
Dan
define the term of domain
Moha
if a>0 then the graph is concave
Angel Reply
if a<0 then the graph is concave blank
Angel
what's a domain
Kamogelo Reply
The set of all values you can use as input into a function su h that the output each time will be defined, meaningful and real.
Spiro
how fast can i understand functions without much difficulty
Joe Reply
what is inequalities
Nathaniel
functions can be understood without a lot of difficulty. Observe the following: f(2) 2x - x 2(2)-2= 2 now observe this: (2,f(2)) ( 2, -2) 2(-x)+2 = -2 -4+2=-2
Dan
what is set?
Kelvin Reply
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
Divya Reply
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
158.5 This number can be developed by using algebra and logarithms. Begin by moving log(2) to the right hand side of the equation like this: t/100 log(2)= log(3) step 1: divide each side by log(2) t/100=1.58496250072 step 2: multiply each side by 100 to isolate t. t=158.49
Dan
what is the importance knowing the graph of circular functions?
Arabella Reply
can get some help basic precalculus
ismail Reply
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
Camalia Reply
can get some help inverse function
ismail
Rectangle coordinate
Asma Reply
how to find for x
Jhon Reply
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
how to find x: 12x = 144 notice how 12 is being multiplied by x. Therefore division is needed to isolate x and whatever we do to one side of the equation we must do to the other. That develops this: x= 144/12 divide 144 by 12 to get x. addition: 12+x= 14 subtract 12 by each side. x =2
Dan
whats a domain
mike Reply
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
Churlene Reply
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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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