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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 the planets on our solar system orbit
cheten Reply
how many Messier objects are there in space
satish Reply
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Richard Reply
what are astronomy
Issan Reply
Astronomy (from Ancient Greek ἀστρονομία (astronomía) 'science that studies the laws of the stars') is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution.
Rafael
vjuvu
Elgoog
what is big bang theory?
Rosemary
what type of activity astronomer do?
Rosemary
No
Richard
the big bang theory is a theory which states that all matter was compressed together in one place the matter got so unstable it exploded releasing All its contents in the form of hydrogen
Roaul
I want to be an astronomer. That's my dream
Astrit
Who named the the whole galaxy?
Shola Reply
solar Univers
GPOWER
what is space
Richard
what is the dark matter
Richard
what are the factors upon which the atmosphere is stratified
Nicholas Reply
is the big bang the sun
Folakemi Reply
no
Sokak
bigbang is the beginning of the universe
Sokak
but thats just a theory
Sokak
nothing will happen, don't worry brother.
Vansh
what does comet means
GANGAIN Reply
these are Rocky substances between mars and jupiter
GANGAIN
Comets are cosmic snowballs of frozen gases , rock and dust that orbit the sun. They are mostly found between the orbits of Venus and Mercury.
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John
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John
hey can anyone guide me abt international astronomy olympiad
sahil
how can we learn right and true ?
Govinda Reply
why the moon is always appear in an elliptical shape
Gatjuol Reply
Because when astroid hit the Earth then a piece of elliptical shape of the earth was separated which is now called moon.
Hemen
what's see level?
lidiya Reply
Did you mean eye sight or sea level
Minal
oh sorry it's sea level
lidiya
according to the theory of astronomers why the moon is always appear in an elliptical orbit?
Gatjuol
hi !!! I am new in astronomy.... I have so many questions in mind .... all of scientists of the word they just give opinion only. but they never think true or false ... i respect all of them... I believes whole universe depending on true ...থিউরি
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Elyana
we're all stars and galaxies a part of sun. how can science prove thx with respect old ancient times picture or books..or anything with respect to present time .but we r a part of that universe
w astronomy and cosmology!
Michele
another theory of universe except big ban
Albash Reply
how was universe born
Asmit Reply
there many theory to born universe but what is the reality of big bang theory to born universe
Asmit
what is the exact value of π?
Nagalakshmi
by big bang
universal
there are many theories regarding this it's on you believe any theory that you think is true ex. eternal inflation theory, oscillation model theory, multiple universe theory the big bang theory etc.
Aarya
I think after Big Bang!
Michele
from where on earth could u observe all the stars during the during the course of an year
Karuna Reply
I think it couldn't possible on earth
Nagalakshmi
in this time i don't Know
Michele
is that so. the question was in the end of this chapter
Karuna
in theory, you could see them all from the equator (though over the course of a year, not at pne time). stars are measured in "declination", which is how far N or S of the equator (90* to -90*). Polaris is the North star, and is ALMOST 90* (+89*). So it would just barely creep over the horizon.
Christopher
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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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