<< Chapter < Page Chapter >> Page >

The process was to divide, multiply, and subtract.

Review of subtraction of polynomials

A very important step in the process of dividing one polynomial by another is subtraction of polynomials. Let’s review the process of subtraction by observing a few examples.

1. Subtract x 2 from x 5 ; that is, find ( x 5 ) ( x 2 ) .

 Since   x 2 is preceded by a minus sign, remove the parentheses, change the sign of each term, then add.

  x 5 ( x 2 ) = x 5 x + 2 3

The result is 3.

2. Subtract x 3 + 3 x 2 from x 3 + 4 x 2 + x 1.

 Since x 3 + 3 x 2 is preceded by a minus sign, remove the parentheses, change the sign of each term, then add.

x 3 + 4 x 2 + x 1 ( x 3 + 3 x 2 ) = x 3 + 4 x 2 + x 1 x 3 3 x 2 x 2 + x 1

The result is x 2 + x 1.

3. Subtract x 2 + 3 x from x 2 + 1.

 We can write x 2 + 1 as x 2 + 0 x + 1.

x 2 + 1 ( x 2 + 3 x ) = x 2 + 0 x + 1 ( x 2 + 3 x ) = x 2 + 0 x + 1 x 2 3 x 3 x + 1

Dividing a polynomial by a polynomial

Now we’ll observe some examples of dividing one polynomial by another. The process is the same as the process used with whole numbers: divide, multiply, subtract, divide, multiply, subtract,....

The division, multiplication, and subtraction take place one term at a time. The process is concluded when the polynomial remainder is of lesser degree than the polynomial divisor.

Sample set b

Perform the division.

x 5 x 2 . We are to divide  x 5  by  x 2.

Long division showing x minus two dividing x minus five with the comment 'Divide x into x' on the right side. This division is not performed completely. See the longdesc for a full description.

1 3 x 2 Thus, x 5 x 2 = 1 3 x 2

Got questions? Get instant answers now!

x 3 + 4 x 2 + x 1 x + 3 . We are to divide  x 3 + 4 x 2 + x 1  by  x + 3.

Long division showing x plus three dividing x cube plus four x square plus x minus one with the comment 'Divide x into x cube' on the right side. This division is not performed completely. See the longdesc for a full description

x 2 + x 2 + 5 x + 3 Thus, x 3 + 4 x 2 + x 1 x + 3 = x 2 + x 2 + 5 x + 3

Got questions? Get instant answers now!

Practice set b

Perform the following divisions.

x + 6 x 1

1 + 7 x 1

Got questions? Get instant answers now!

x 2 + 2 x + 5 x + 3

x 1 + 8 x + 3

Got questions? Get instant answers now!

x 3 + x 2 x 2 x + 8

x 2 7 x + 55 442 x + 8

Got questions? Get instant answers now!

x 3 + x 2 3 x + 1 x 2 + 4 x 5

x 3 + 14 x 14 x 2 + 4 x 5 = x 3 + 14 x + 5

Got questions? Get instant answers now!

Sample set c

Divide  2 x 3 4 x + 1 by  x + 6. 2 x 3 4 x + 1 x + 6 Notice that the  x 2  term in the numerator is missing .  We can avoid any confusion by writing 2 x 3 + 0 x 2 4 x + 1 x + 6 Divide, multiply, and subtract .

Steps of long division showing the quantity x plus six dividing the quantity two x cubed plus zero x squared minus four x minus plus one. See the longdesc for a full description

2 x 3 4 x + 1 x + 6 = 2 x 3 12 x + 68 407 x + 6

Got questions? Get instant answers now!

Practice set c

Perform the following divisions.

x 2 3 x + 2

x 2 + 1 x + 2

Got questions? Get instant answers now!

4 x 2 1 x 3

4 x + 12 + 35 x 3

Got questions? Get instant answers now!

x 3 + 2 x + 2 x 2

x 2 + 2 x + 6 + 14 x 2

Got questions? Get instant answers now!

6 x 3 + 5 x 2 1 2 x + 3

3 x 2 2 x + 3 10 2 x + 3

Got questions? Get instant answers now!

Exercises

For the following problems, perform the divisions.

8 y 4 4

2 y + 1

Got questions? Get instant answers now!

3 x 2 6 x 3

x ( x 2 )

Got questions? Get instant answers now!

6 x 3 + 2 x 2 + 8 x 2 x

3 x 2 + x + 4

Got questions? Get instant answers now!

26 y 3 + 13 y 2 + 39 y 13 y

Got questions? Get instant answers now!

a 2 b 2 + 4 a 2 b + 6 a b 2 10 a b a b

a b + 4 a + 6 b 10

Got questions? Get instant answers now!

7 x 3 y + 8 x 2 y 3 + 3 x y 4 4 x y x y

Got questions? Get instant answers now!

5 x 3 y 3 15 x 2 y 2 + 20 x y 5 x y

x 2 y 2 + 3 x y 4

Got questions? Get instant answers now!

4 a 2 b 3 8 a b 4 + 12 a b 2 2 a b 2

Got questions? Get instant answers now!

6 a 2 y 2 + 12 a 2 y + 18 a 2 24 a 2

1 4 y 2 + 1 2 y + 3 4

Got questions? Get instant answers now!

3 c 3 y 3 + 99 c 3 y 4 12 c 3 y 5 3 c 3 y 3

Got questions? Get instant answers now!

16 a x 2 20 a x 3 + 24 a x 4 6 a 4

8 x 2 10 x 3 + 12 x 4 3 a 3 or 12 x 4 10 x 3 + 8 x 2 3 a 3

Got questions? Get instant answers now!

21 a y 3 18 a y 2 15 a y 6 a y 2

Got questions? Get instant answers now!

14 b 2 c 2 + 21 b 3 c 3 28 c 3 7 a 2 c 3

2 b 2 3 b 3 c + 4 c a 2 c

Got questions? Get instant answers now!

30 a 2 b 4 35 a 2 b 3 25 a 2 5 b 3

Got questions? Get instant answers now!

x + 6 x 2

1 + 8 x 2

Got questions? Get instant answers now!

x 2 x + 4 x + 2

x 3 + 10 x + 2

Got questions? Get instant answers now!

x 2 x + 3 x + 1

x 2 + 5 x + 1

Got questions? Get instant answers now!

x 2 2 x + 1

x 1 1 x + 1

Got questions? Get instant answers now!

y 2 + 4 y + 2

y 2 + 8 y + 2

Got questions? Get instant answers now!

x 3 1 x + 1

x 2 x + 1 2 x + 1

Got questions? Get instant answers now!

x 3 1 x 1

x 2 + x + 1

Got questions? Get instant answers now!

x 3 + 3 x 2 + x 2 x 2

x 2 + 5 x + 11 + 20 x 2

Got questions? Get instant answers now!

a 3 + 2 a 2 a + 1 a 3

Got questions? Get instant answers now!

a 3 + a + 6 a 1

a 2 + a + 2 + 8 a 1

Got questions? Get instant answers now!

y 3 + 3 y 2 + 4 y + 2

y 2 + y 2 + 8 y + 2

Got questions? Get instant answers now!

y 3 + 5 y 2 3 y 1

Got questions? Get instant answers now!

x 2 x 6 x 2 2 x 3

1 + 1 x + 1

Got questions? Get instant answers now!

a 2 + 5 a + 4 a 2 a 2

Got questions? Get instant answers now!

2 y 2 + 5 y + 3 y 2 3 y 4

2 + 11 y 4

Got questions? Get instant answers now!

3 a 2 + 4 a 4 a 2 + 3 a + 3

Got questions? Get instant answers now!

2 x 2 x + 4 2 x 1

x + 4 2 x 1

Got questions? Get instant answers now!

6 x 2 + 8 x 1 3 x + 4

2 x 1 3 x + 4

Got questions? Get instant answers now!

20 y 2 + 15 y 4 4 y + 3

Got questions? Get instant answers now!

4 x 3 + 4 x 2 3 x 2 2 x 1

2 x 2 + 3 x 2 2 x 1

Got questions? Get instant answers now!

9 a 3 18 a 2 + 8 a 1 3 a 2

Got questions? Get instant answers now!

4 x 4 4 x 3 + 2 x 2 2 x 1 x 1

4 x 3 + 2 x 1 x 1

Got questions? Get instant answers now!

3 y 4 + 9 y 3 2 y 2 6 y + 4 y + 3

Got questions? Get instant answers now!

3 y 2 + 3 y + 5 y 2 + y + 1

3 + 2 y 2 + y + 1

Got questions? Get instant answers now!

2 a 2 + 4 a + 1 a 2 + 2 a + 3

Got questions? Get instant answers now!

8 z 6 4 z 5 8 z 4 + 8 z 3 + 3 z 2 14 z 2 z 3

4 z 5 + 4 z 4 + 2 z 3 + 7 z 2 + 12 z + 11 + 33 2 z 3

Got questions? Get instant answers now!

9 a 7 + 15 a 6 + 4 a 5 3 a 4 a 3 + 12 a 2 + a 5 3 a + 1

Got questions? Get instant answers now!

( 2 x 5 + 5 x 4 1 ) ÷ ( 2 x + 5 )

x 4 1 2 x + 5

Got questions? Get instant answers now!

( 6 a 4 2 a 3 3 a 2 + a + 4 ) ÷ ( 3 a 1 )

Got questions? Get instant answers now!

Exercises for review

( [link] ) Find the product. x 2 + 2 x 8 x 2 9 · 2 x + 6 4 x 8 .

x + 4 2 ( x 3 )

Got questions? Get instant answers now!

( [link] ) Find the sum. x 7 x + 5 + x + 4 x 2 .

Got questions? Get instant answers now!

( [link] ) Solve the equation 1 x + 3 + 1 x 3 = 1 x 2 9 .

x = 1 2

Got questions? Get instant answers now!

( [link] ) When the same number is subtracted from both the numerator and denominator of 3 10 , the result is 1 8 . What is the number that is subtracted?

Got questions? Get instant answers now!

( [link] ) Simplify 1 x + 5 4 x 2 25 .

x 5 4

Got questions? Get instant answers now!

Questions & Answers

what does mean opportunity cost?
Aster Reply
what is poetive effect of population growth
Solomon Reply
what is inflation
Nasir Reply
what is demand
Eleni
what is economics
IMLAN Reply
economics theory describes individual behavior as the result of a process of optimization under constraints the objective to be reached being determined by
Kalkidan
Economics is a branch of social science that deal with How to wise use of resource ,s
Kassie
need
WARKISA
Economic Needs: In economics, needs are goods or services that are necessary for maintaining a certain standard of living. This includes things like healthcare, education, and transportation.
Kalkidan
What is demand and supply
EMPEROR Reply
deman means?
Alex
what is supply?
Alex
ex play supply?
Alex
Money market is a branch or segment of financial market where short-term debt instruments are traded upon. The instruments in this market includes Treasury bills, Bonds, Commercial Papers, Call money among other.
murana Reply
good
Kayode
what is money market
umar Reply
Examine the distinction between theory of comparative cost Advantage and theory of factor proportion
Fatima Reply
What is inflation
Bright Reply
a general and ongoing rise in the level of prices in an economy
AI-Robot
What are the factors that affect demand for a commodity
Florence Reply
price
Kenu
differentiate between demand and supply giving examples
Lambiv Reply
differentiated between demand and supply using examples
Lambiv
what is labour ?
Lambiv
how will I do?
Venny Reply
how is the graph works?I don't fully understand
Rezat Reply
information
Eliyee
devaluation
Eliyee
t
WARKISA
hi guys good evening to all
Lambiv
multiple choice question
Aster Reply
appreciation
Eliyee
explain perfect market
Lindiwe Reply
In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,
Ezea
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elementary algebra' conversation and receive update notifications?

Ask