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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

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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

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Practice set b

Perform the following divisions.

x + 6 x 1

1 + 7 x 1

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x 2 + 2 x + 5 x + 3

x 1 + 8 x + 3

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x 3 + x 2 x 2 x + 8

x 2 7 x + 55 442 x + 8

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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

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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

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Practice set c

Perform the following divisions.

x 2 3 x + 2

x 2 + 1 x + 2

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4 x 2 1 x 3

4 x + 12 + 35 x 3

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x 3 + 2 x + 2 x 2

x 2 + 2 x + 6 + 14 x 2

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

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

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Exercises

For the following problems, perform the divisions.

8 y 4 4

2 y + 1

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3 x 2 6 x 3

x ( x 2 )

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

3 x 2 + x + 4

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

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

a b + 4 a + 6 b 10

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

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

x 2 y 2 + 3 x y 4

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

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

1 4 y 2 + 1 2 y + 3 4

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

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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

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

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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

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

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x + 6 x 2

1 + 8 x 2

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x 2 x + 4 x + 2

x 3 + 10 x + 2

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x 2 x + 3 x + 1

x 2 + 5 x + 1

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x 2 2 x + 1

x 1 1 x + 1

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y 2 + 4 y + 2

y 2 + 8 y + 2

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x 3 1 x + 1

x 2 x + 1 2 x + 1

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x 3 1 x 1

x 2 + x + 1

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x 3 + 3 x 2 + x 2 x 2

x 2 + 5 x + 11 + 20 x 2

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a 3 + 2 a 2 a + 1 a 3

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a 3 + a + 6 a 1

a 2 + a + 2 + 8 a 1

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y 3 + 3 y 2 + 4 y + 2

y 2 + y 2 + 8 y + 2

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y 3 + 5 y 2 3 y 1

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x 2 x 6 x 2 2 x 3

1 + 1 x + 1

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a 2 + 5 a + 4 a 2 a 2

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

2 + 11 y 4

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

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2 x 2 x + 4 2 x 1

x + 4 2 x 1

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6 x 2 + 8 x 1 3 x + 4

2 x 1 3 x + 4

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20 y 2 + 15 y 4 4 y + 3

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

2 x 2 + 3 x 2 2 x 1

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

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

4 x 3 + 2 x 1 x 1

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

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

3 + 2 y 2 + y + 1

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

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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

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

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( 2 x 5 + 5 x 4 1 ) ÷ ( 2 x + 5 )

x 4 1 2 x + 5

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

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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 )

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

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

x = 1 2

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( [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?

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

x 5 4

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Questions & Answers

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research.net
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Introduction about quantum dots in nanotechnology
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there is no specific books for beginners but there is book called principle of nanotechnology
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are you nano engineer ?
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fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
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what is the actual application of fullerenes nowadays?
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That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
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Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
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is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
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of graphene you mean?
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or in general
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in general
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Graphene has a hexagonal structure
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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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