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3 x 2 75

3 ( x 5 ) ( x 5 )

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a 3 b 4 m a m 3 n 2

a m ( a b 2 + m n ) ( a b 2 m n )

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Fundamental rules of factoring

There are two fundamental rules that we follow when factoring:

    Fundamental rules of factoring

  1. Factor out all common monomials first.
  2. Factor completely.

Sample set b

Factor each binomial completely.

4 a 8 b - 36 b 5 . Factor out the common factor 4b. 4 b ( a 8 - 9 b 4 )

Now we can see a difference of two squares, whereas in the original polynomial we could not. We’ll complete our factorization by factoring the difference of two squares.

4 a 8 b - 36 b 5 = 4 b ( a 8 - 9 b 4 ) = 4 b ( a 4 + 3 b 2 ) ( a 4 - 3 b 2 )

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x 16 - y 8 . Factor this difference of two squares . x 16 - y 8 = ( x 8 + y 4 ) Sum of two squares Does not factor ( x 8 - y 4 )  Difference of two squares Factor it! = ( x 8 + y 4 ) ( x 4 + y 2 ) ( x 4 - y 2 ) Factor again! = ( x 8 + y 4 ) ( x 4 + y 2 ) ( x 2 + y ) ( x 2 - y )

Finally, the factorization is complete.

These types of products appear from time to time, so be aware that you may have to factor more than once.

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Practice set b

Factor each binomial completely.

m 4 n 4

( m 2 + n 2 ) ( m n ) ( m + n )

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16 y 8 1

( 4 y 4 + 1 ) ( 2 y 2 + 1 ) ( 2 y 2 1 )

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Perfect square trinomials

Recall the process of squaring a binomial.

( a b ) 2 a 2 2 a b b 2 ( a b ) 2 a 2 2 a b b 2

Our Method Is We Notice
Square the first term. The first term of the product should be a perfect square.
Take the product of the two terms and double it. The middle term of the product should be divisible by 2 (since it’s multiplied by 2).
Square the last term. The last term of the product should be a perfect square.

Perfect square trinomials always factor as the square of a binomial.

To recognize a perfect square trinomial, look for the following features:

  1. The first and last terms are perfect squares.
  2. The middle term is divisible by 2, and if we divide the middle term in half (the opposite of doubling it), we will get the product of the terms that when squared produce the first and last terms.

In other words, factoring a perfect square trinomial amounts to finding the terms that, when squared, produce the first and last terms of the trinomial, and substituting into one of the formula

a 2 + 2 a b + b 2 = ( a + b ) 2 a 2 - 2 a b + b 2 = ( a - b ) 2

Sample set c

Factor each perfect square trinomial.

x 2 + 6 x + 9 . This expression is a perfect square trinomial. The x 2 and 9 are perfect squares.

The terms that when squared produce x 2 and 9 are x and 3, respectively.

The middle term is divisible by 2, and 6 x 2 3 x . The 3 x is the product of x and 3, which are the terms that produce the perfect squares.

x 2 + 6 x + 9 = ( x + 3 ) 2

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x 4 - 10 x 2 y 3 + 25 y 6 . This expression is a perfect square trinomial. The x 4 and 25 y 6 are both perfect squares. The terms that when squared produce x 4 and 25 y 6 are x 2 and 5 y 3 , respectively.

The middle term - 10 x 2 y 3 is divisible by 2 . In fact, - 10 x 2 y 3 2 = - 5 x 2 y 3 . Thus,

x 4 - 10 x 2 y 3 + 25 y 6 = ( x 2 - 5 y 3 ) 2

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x 2 + 10 x + 16 . This expression is not a perfect square trinomial. Although the middle term is divisible by 2 , ? 10 x 2 = 5 x , the 5 and x are not the terms that when squared produce the first and last terms. (This expression would be a perfect square trinomial if the middle term were 8 x .)

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4 a 4 + 32 a 2 b - 64 b 2 . This expression is not a perfect square trinomial since the last term - 64 b 2 is not a perfect square (since any quantity squared is always positive or zero and never negative).

Thus, 4 a 4 + 32 a 2 b - 64 b 2 cannot be factored using this method.

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Practice set c

Factor, if possible, the following trinomials.

m 2 - 8 m + 16

( m - 4 ) 2

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k 2 + 10 k + 25

( k + 5 ) 2

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4 a 2 + 12 a + 9

( 2 a + 3 ) 2

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9 x 2 - 24 x y + 16 y 2

( 3 x - 4 y ) 2

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2 w 3 z + 16 w 2 z 2 + 32 w z 3

2 w z ( w + 4 z ) 2

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x 2 + 12 x + 49

not possible

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Exercises

For the following problems, factor the binomials.

a 2 9

( a + 3 ) ( a 3 )

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x 2 16

( x + 4 ) ( x 4 )

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a 2 100

( a + 10 ) ( a 10 )

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4 a 2 64

4 ( a + 4 ) ( a 4 )

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3 x 2 27

3 ( x + 3 ) ( x 3 )

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4 a 2 25

( 2 a + 5 ) ( 2 a 5 )

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36 y 2 25

( 6 y + 5 ) ( 6 y 5 )

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12 a 2 75

3 ( 2 a + 5 ) ( 2 a 5 )

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8 y 2 50

2 ( 2 y + 5 ) ( 2 y 5 )

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x 2 y 2 25

( x y + 5 ) ( x y 5 )

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x 4 y 4 9 a 2

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

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4 a 2 b 2 9 b 2

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

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a 2 b 2

( a + b ) ( a b )

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x 4 y 4

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

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a 8 y 2

( a 4 + y ) ( a 4 y )

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b 6 x 4

( b 3 + x 2 ) ( b 3 x 2 )

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25 a 2

( 5 + a ) ( 5 a )

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100 36 b 4

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

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x 4 16

( x 2 + 4 ) ( x + 2 ) ( x 2 )

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a 4 b 4

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

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x 12 y 12

( x 6 + x 6 ) ( x 3 + y 3 ) ( x 3 y 3 )

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a 3 c 2 25 a c 2

a c 2 ( a + 5 ) ( a 5 )

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a 4 b 4 c 2 d 2 36 x 2 y 2

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49 x 2 y 4 z 6 64 a 4 b 2 c 8 d 10

( 7 x y 2 z 3 + 8 a 2 b c 4 d 5 ) ( 7 x y 2 z 3 8 a 2 b c 4 d 5 )

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For the following problems, factor, if possible, the trinomials.

y 2 20 y 100

( y + 10 ) 2

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x 2 10 x 25

( x 5 ) 2

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a 2 24 a 144

( a 12 ) 2

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9 x 2 6 x 1

( 3 x + 1 ) 2

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16 a 2 24 a 9

( 4 a 3 ) 2

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9 x 2 6 x y y 2

( 3 x + y ) 2

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36 a 2 60 a b 25 b 2

( 6 a + 5 b ) 2

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12 a 2 60 a 75

3 ( 2 a 5 ) 2

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32 x 2 16 x 2

2 ( 4 x + 1 ) 2

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4 a 2 a 9

not factorable

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x 5 8 x 4 16 x 3

x 3 ( x + 4 ) 2

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12 a 3 b - 48 a 2 b 2 + 48 a b 3

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Exercises for review

( [link] ) Factor ( m 3 ) x ( m 3 ) y .

( m 3 ) ( x y )

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( [link] ) Factor 8 x m 16 x n 3 y m 6 y n by grouping.

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Questions & Answers

how does Neisseria cause meningitis
Nyibol Reply
what is microbiologist
Muhammad Reply
what is errata
Muhammad
is the branch of biology that deals with the study of microorganisms.
Ntefuni Reply
What is microbiology
Mercy Reply
studies of microbes
Louisiaste
when we takee the specimen which lumbar,spin,
Ziyad Reply
How bacteria create energy to survive?
Muhamad Reply
Bacteria doesn't produce energy they are dependent upon their substrate in case of lack of nutrients they are able to make spores which helps them to sustain in harsh environments
_Adnan
But not all bacteria make spores, l mean Eukaryotic cells have Mitochondria which acts as powerhouse for them, since bacteria don't have it, what is the substitution for it?
Muhamad
they make spores
Louisiaste
what is sporadic nd endemic, epidemic
Aminu Reply
the significance of food webs for disease transmission
Abreham
food webs brings about an infection as an individual depends on number of diseased foods or carriers dully.
Mark
explain assimilatory nitrate reduction
Esinniobiwa Reply
Assimilatory nitrate reduction is a process that occurs in some microorganisms, such as bacteria and archaea, in which nitrate (NO3-) is reduced to nitrite (NO2-), and then further reduced to ammonia (NH3).
Elkana
This process is called assimilatory nitrate reduction because the nitrogen that is produced is incorporated in the cells of microorganisms where it can be used in the synthesis of amino acids and other nitrogen products
Elkana
Examples of thermophilic organisms
Shu Reply
Give Examples of thermophilic organisms
Shu
advantages of normal Flora to the host
Micheal Reply
Prevent foreign microbes to the host
Abubakar
they provide healthier benefits to their hosts
ayesha
They are friends to host only when Host immune system is strong and become enemies when the host immune system is weakened . very bad relationship!
Mark
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faisal Reply
cell is the smallest unit of life
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cell is the smallest unit of life
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ok
Innocent
cell is the structural and functional unit of life
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is the fundamental units of Life
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Micheal Reply
There are nothing like emergency disease but there are some common medical emergency which can occur simultaneously like Bleeding,heart attack,Breathing difficulties,severe pain heart stock.Hope you will get my point .Have a nice day ❣️
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I think infection prevention and control is the avoidance of all things we do that gives out break of infections and promotion of health practices that promote life
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ESTHER Reply
Many sites of the body have it Skin Nasal cavity Oral cavity Gastro intestinal tract
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skin
Asiina
skin,Oral,Nasal,GIt
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all
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by fussion
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part of a tissue or an organ being wounded or bruised.
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what term is used to name and classify microorganisms?
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Binomial nomenclature
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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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