<< 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 is defense mechanism
Chinaza Reply
what is defense mechanisms
Chinaza
I'm interested in biological psychology and cognitive psychology
Tanya Reply
what does preconceived mean
sammie Reply
physiological Psychology
Nwosu Reply
How can I develope my cognitive domain
Amanyire Reply
why is communication effective
Dakolo Reply
Communication is effective because it allows individuals to share ideas, thoughts, and information with others.
effective communication can lead to improved outcomes in various settings, including personal relationships, business environments, and educational settings. By communicating effectively, individuals can negotiate effectively, solve problems collaboratively, and work towards common goals.
it starts up serve and return practice/assessments.it helps find voice talking therapy also assessments through relaxed conversation.
miss
Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment
Wekolamo Reply
please i need answer
Wekolamo
because it helps many people around the world to understand how to interact with other people and understand them well, for example at work (job).
Manix Reply
Agreed 👍 There are many parts of our brains and behaviors, we really need to get to know. Blessings for everyone and happy Sunday!
ARC
A child is a member of community not society elucidate ?
JESSY Reply
Isn't practices worldwide, be it psychology, be it science. isn't much just a false belief of control over something the mind cannot truly comprehend?
Simon Reply
compare and contrast skinner's perspective on personality development on freud
namakula Reply
Skinner skipped the whole unconscious phenomenon and rather emphasized on classical conditioning
war
explain how nature and nurture affect the development and later the productivity of an individual.
Amesalu Reply
nature is an hereditary factor while nurture is an environmental factor which constitute an individual personality. so if an individual's parent has a deviant behavior and was also brought up in an deviant environment, observation of the behavior and the inborn trait we make the individual deviant.
Samuel
I am taking this course because I am hoping that I could somehow learn more about my chosen field of interest and due to the fact that being a PsyD really ignites my passion as an individual the more I hope to learn about developing and literally explore the complexity of my critical thinking skills
Zyryn Reply
good👍
Jonathan
and having a good philosophy of the world is like a sandwich and a peanut butter 👍
Jonathan
generally amnesi how long yrs memory loss
Kelu Reply
interpersonal relationships
Abdulfatai Reply
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