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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

Discuss the differences between taste and flavor, including how other sensory inputs contribute to our  perception of flavor.
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While taste primarily relies on our taste buds, flavor involves a complex interplay between taste and aroma
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to drain extracellular fluid all over the body.
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The lymphatic system plays several crucial roles in the human body, functioning as a key component of the immune system and contributing to the maintenance of fluid balance. Its main functions include: 1. Immune Response: The lymphatic system produces and transports lymphocytes, which are a type of
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Anatomy is the identification and description of the structures of living things
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Anatomy is the study of the structure of the body, while physiology is the study of the function of the body. Anatomy looks at the body's organs and systems, while physiology looks at how those organs and systems work together to keep the body functioning.
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Enzymes are proteins that help speed up chemical reactions in our bodies. Enzymes are essential for digestion, liver function and much more. Too much or too little of a certain enzyme can cause health problems
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it is because of the enzyme that the stomach produce that help the stomach from the damaging effect of HCL
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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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