<< 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

Why is b in the answer
Dahsolar Reply
how do you work it out?
Brad Reply
answer
Ernest
heheheehe
Nitin
(Pcos∅+qsin∅)/(pcos∅-psin∅)
John Reply
how to do that?
Rosemary Reply
what is it about?
Amoah
how to answer the activity
Chabelita Reply
how to solve the activity
Chabelita
solve for X,,4^X-6(2^)-16=0
Alieu Reply
x4xminus 2
Lominate
sobhan Singh jina uniwarcity tignomatry ka long answers tile questions
harish Reply
t he silly nut company makes two mixtures of nuts: mixture a and mixture b. a pound of mixture a contains 12 oz of peanuts, 3 oz of almonds and 1 oz of cashews and sells for $4. a pound of mixture b contains 12 oz of peanuts, 2 oz of almonds and 2 oz of cashews and sells for $5. the company has 1080
ZAHRO Reply
If  , , are the roots of the equation 3 2 0, x px qx r     Find the value of 1  .
Swetha Reply
Parts of a pole were painted red, blue and yellow. 3/5 of the pole was red and 7/8 was painted blue. What part was painted yellow?
Patrick Reply
Parts of the pole was painted red, blue and yellow. 3 /5 of the pole was red and 7 /8 was painted blue. What part was painted yellow?
Patrick
how I can simplify algebraic expressions
Katleho Reply
Lairene and Mae are joking that their combined ages equal Sam’s age. If Lairene is twice Mae’s age and Sam is 69 yrs old, what are Lairene’s and Mae’s ages?
Mary Reply
23yrs
Yeboah
lairenea's age is 23yrs
ACKA
hy
Katleho
Ello everyone
Katleho
Laurene is 46 yrs and Mae is 23 is
Solomon
hey people
christopher
age does not matter
christopher
solve for X, 4^x-6(2*)-16=0
Alieu
prove`x^3-3x-2cosA=0 (-π<A<=π
Mayank Reply
create a lesson plan about this lesson
Rose Reply
Excusme but what are you wrot?
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