<< 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

summarize halerambos & holbon
David Reply
the Three stages of Auguste Comte
Clementina Reply
what are agents of socialization
Antonio Reply
sociology of education
Nuhu Reply
definition of sociology of education
Nuhu
what is culture
Abdulrahim Reply
shared beliefs, values, and practices
AI-Robot
What are the two type of scientific method
ogunniran Reply
I'm willing to join you
Aceng Reply
what are the scientific method of sociology
Man
what is socialization
ogunniran Reply
the process wherein people come to understand societal norms and expectations, to accept society's beliefs, and to be aware of societal values
AI-Robot
scientific method in doing research
ogunniran
defimition of sickness in afica
Anita
Cosmology
ogunniran
Hmmm
ogunniran
list and explain the terms that found in society
REMMY Reply
list and explain the terms that found in society
Mukhtar
what are the agents of socialization
Antonio
Family Peer group Institution
Abdulwajud
I mean the definition
Antonio
ways of perceived deviance indifferent society
Naomi Reply
reasons of joining groups
SAM
to bring development to the nation at large
Hyellafiya
entails of consultative and consensus building from others
Gadama
World first Sociologist?
Abu
What is evolutionary model
Muhammad Reply
Evolution models refer to mathematical and computational representations of the processes involved in biological evolution. These models aim to simulate and understand how species change over time through mechanisms such as natural selection, genetic drift, and mutation. Evolutionary models can be u
faruk
what are the modern trends in religious behaviours
Selekeye Reply
what are social norms
Daniel Reply
shared standards of acceptable behavior by the group or appropriate behavior in a particular institution or those behaviors that are acceptable in a society
Lucius
that is how i understood it
Lucius
examples of societal norms
Diamond
Discuss the characteristics of the research located within positivist and the interpretivist paradigm
Tariro Reply
what is Industrialisation
Selekeye Reply
industrialization
Angelo
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