<< Chapter < Page Chapter >> Page >
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:

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 = ( 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

the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
1+cos²A/cos²A=2cosec²A-1
Ramesh Reply
test for convergence the series 1+x/2+2!/9x3
success Reply
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
Lhorren Reply
100 meters
Kuldeep
Find that number sum and product of all the divisors of 360
jancy Reply
answer
Ajith
exponential series
Naveen
what is subgroup
Purshotam Reply
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
Macmillan Reply
e power cos hyperbolic (x+iy)
Vinay Reply
10y
Michael
tan hyperbolic inverse (x+iy)=alpha +i bita
Payal Reply
prove that cos(π/6-a)*cos(π/3+b)-sin(π/6-a)*sin(π/3+b)=sin(a-b)
Tejas Reply
why {2kπ} union {kπ}={kπ}?
Huy Reply
why is {2kπ} union {kπ}={kπ}? when k belong to integer
Huy
if 9 sin theta + 40 cos theta = 41,prove that:41 cos theta = 41
Trilochan Reply
what is complex numbers
Ayushi Reply
Please you teach
Dua
Yes
ahmed
Thank you
Dua
give me treganamentry question
Anshuman Reply
Solve 2cos x + 3sin x = 0.5
shobana Reply
Practice Key Terms 2

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask