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Sample set b

Factor 24 x 2 41 x + 12 .

Factor the first and last terms.

24 x 2 12
24 x , x 12 , 1
12 x , 2 x 6 , 2
8 x , 3 x 4 , 3
6 x , 4 x

Rather than starting with the 24 x , x and 12 , 1 , pick some intermediate values, 8 x and 3 x , the 6 x and 4 x , or the 6 and 2 , or the 4 and 3 .

24 x 2 41 x + 12 = ( 8 x 3 ) ( 3 x 4 )

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Practice set b

Factor 48 x 2 + 22 x - 15 .

( 6 x + 5 ) ( 8 x 3 )

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Factor 54 y 2 + 39 y w 28 w 2 .

( 9 y 4 w ) ( 6 y + 7 w )

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The collect and discard method of factoring a x 2 + b x + c

Collect and discard method

Consider the polynomial 6 x 2 + x 12 . We begin by identifying a and c . In this case, a = 6 and c = 12 . We start out as we would with a = 1 .

6 x 2 + x 12 : ( 6 x ) ( 6 x )

Now, compute a c .

a c = ( 6 ) ( 12 ) = 72

Find the factors of 72 that add to 1, the coefficient of x , the linear term. The factors are 9 and 8 . Include these factors in the parentheses.

6 x 2 + x 12 : ( 6 x + 9 ) ( 6 x 8 )

But we have included too much. We must eliminate the surplus. Factor each parentheses.

6 x 2 + x 12 : 3 ( 2 x + 3 ) 2 ( 3 x 4 )

Discard the factors that multiply to a = 6 . In this case, 3 and 2. We are left with the proper factorization.

6 x 2 + x 12 = ( 2 x + 3 ) ( 3 x 4 )

Sample set c

Factor 10 x 2 + 23 x 5 .

Identify a = 10 and b = 5 .

10 x 2 + 23 x 5 ; ( 10 x ) ( 10 x )

Compute

a c = ( 10 ) ( 5 ) = 50

Find the factors of 50 that add to + 23 , the coefficient of x , the linear term. The factors are 25 and 2 . Place these numbers into the parentheses.

10 x 2 + 23 x 5 : ( 10 x + 25 ) ( 10 x 2 )

We have collected too much. Factor each set of parentheses and eliminate the surplus.

10 x 2 + 23 x 5 : ( 5 ) ( 2 x + 5 ) ( 2 ) ( 5 x 1 )

Discard the factors that multiply to a = 10 . In this case, 5 and 2.

10 x 2 + 23 x 5 = ( 2 x + 5 ) ( 5 x 1 )

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Factor 8 x 2 30 x 27 .

Identify a = 8 and c = 27 .

8 x 2 30 x 27 : ( 8 x ) ( 8 x )

Compute

a c = ( 8 ) ( 27 ) = 216

Find the factors of 216 that add to 30 , the coefficient of x , the linear term. This requires some thought. The factors are 36 and 6. Place these numbers into the parentheses.

8 x 2 30 x 27 : ( 8 x 36 ) ( 8 x + 6 )

We have collected too much. Factor each set of parentheses and eliminate the surplus.

8 x 2 30 x 27 : ( 4 ) ( 2 x 9 ) ( 2 ) ( 4 x + 3 )

Discard the factors that multiply to a = 8 . In this case, 4 and 2.

8 x 2 30 x 27 = ( 2 x 9 ) ( 4 x + 3 )

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Factor 18 x 2 5 x y 2 y 2 .

Identify a = 18 and c = 2 .

18 x 2 5 x y 2 y 2 : ( 18 x ) ( 18 x )

Compute

a c = ( 18 ) ( 2 ) = 36

Find the factors of 36 that add to 5 , the coefficient of x y . In this case, 9 and 4. Place these numbers into the parentheses, affixing y to each.

18 x 2 5 x y 2 y 2 : ( 18 x 9 y ) ( 18 x + 4 y )

We have collected too much. Factor each set of parentheses and eliminate the surplus.

18 x 2 5 x y 2 y 2 : ( 9 ) ( 2 x y ) ( 2 ) ( 9 x + 2 y )

Discard the factors that multiply to a = 18 . In this case, 9 and 4.

18 x 2 5 x y 2 y 2 = ( 2 x y ) ( 9 x + 2 y )

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Practice set c

Factor 6 x 2 + 7 x 3 .

( 3 x 1 ) ( 2 x + 3 )

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Factor 14 x 2 31 x 10 .

( 7 x + 2 ) ( 2 x 5 )

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Factor 48 x 2 + 22 x 15 .

( 6 x + 5 ) ( 8 x 3 )

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Factor 10 x 2 23 x w + 12 w 2 .

( 5 x 4 w ) ( 2 x 3 w )

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Exercises

Factor the following problems, if possible.

x 2 + 3 x + 2

( x + 2 ) ( x + 1 )

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2 x 2 + 7 x + 5

( 2 x + 5 ) ( x + 1 )

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2 x 2 + 11 x + 12

( 2 x + 3 ) ( x + 4 )

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3 x 2 x 4

( 3 x 4 ) ( x + 1 )

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4 x 2 + 8 x 21

( 2 x 3 ) ( 2 x + 7 )

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9 a 2 7 a + 2

not factorable

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16 y 2 - 26 y + 3

( 8 y 1 ) ( 2 y 3 )

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10 x 2 + 29 x + 10

( 5 x + 2 ) ( 2 x + 5 )

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81 a 2 + 19 a + 2

not factorable

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24 x 2 34 x + 5

( 6 x 1 ) ( 4 x 5 )

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24 x 2 + 26 x 5

( 6 x 1 ) ( 4 x + 5 )

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6 x 2 + 5 x y + y 2

( 3 x + y ) ( 2 x + y )

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For the following problems, the given trinomial occurs when solving the corresponding applied problem. Factor each trinomial. You do not need to solve the problem.

5 r 2 24 r 5 .

It takes 5 hours to paddle a boat 12 miles downstream and then back. The current flows at the rate of 1 mile per hour. At what rate was the boat paddled?

( 5 r + 1 ) ( r 5 )

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x 2 + 5 x 84 .

The length of a rectangle is 5 inches more than the width of the rectangle. If the area of the rectangle is 84 square inches, what are the length and width of the rectangle?

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x 2 + 24 x 145 .

A square measures 12 inches on each side. Another square is to be drawn around this square in such a way that the total area is 289 square inches. What is the distance from the edge of the smaller square to the edge of the larger square? (The two squares have the same center.)

( x + 29 ) ( x 5 )

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x 2 + 8 x 20 .

A woman wishes to construct a rectangular box that is open at the top. She wishes it to be 4 inches high and have a rectangular base whose length is three times the width. The material used for the base costs $2 per square inch, and the material used for the sides costs $1.50 per square inch. The woman will spend exactly $120 for materials. Find the dimension of the box (length of the base, width of the base, and height).

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For the following problems, factor the trinomials if possible.

16 x 2 8 x y 3 y 2

( 4 x + y ) ( 4 x 3 y )

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12 a 2 + 7 a b + 12 b 2

not factorable

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8 a 2 + 10 a b 6 b 2

2 ( 4 a 2 + 5 a b 3 b 2 )

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12 b 4 + 30 b 2 a + 12 a 2

6 ( 2 b 2 + a ) ( b 2 + 2 a )

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30 a 4 b 4 3 a 2 b 2 6 c 2

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3 a 6 3 a 3 b 2 18 b 4

3 ( a 3 + 2 b 2 ) ( a 3 3 b 2 )

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20 a 2 b 2 + 2 a b c 2 6 a 2 c 4

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14 a 2 z 2 40 a 3 z 2 46 a 4 z 2

2 a 2 z 2 ( 7 20 a 23 a 2 )  or  2 a 2 z 2 ( 23 a 2 + 20 a 7 )

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Exercises for review

( [link] ) Simplify ( a 3 b 6 ) 4 .

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( [link] ) Find the product. x 2 ( x 3 ) ( x + 4 ) .

x 4 + x 3 12 x 2

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( [link] ) Find the product. ( 5 m 3 n ) 2 .

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( [link] ) Solve the equation 5 ( 2 x 1 ) 4 ( x + 7 ) = 0 .

x = 11 2

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( [link] ) Factor x 5 8 x 4 + 7 x 3 .

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Questions & Answers

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s. Reply
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so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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s. Reply
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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I start with an easy one. carbon nanotubes woven into a long filament like a string
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preparation of nanomaterial
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Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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good afternoon madam
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In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
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anybody can imagine what will be happen after 100 years from now in nano tech world
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after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
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name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
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not now but maybe in future only AgNP maybe any other nanomaterials
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this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
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Prasenjit Reply
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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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