<< Chapter < Page Chapter >> Page >

3 x 2 75

3 ( x 5 ) ( x 5 )

Got questions? Get instant answers now!

a 3 b 4 m a m 3 n 2

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

Got questions? Get instant answers now!

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 )

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

Practice set b

Factor each binomial completely.

m 4 n 4

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

Got questions? Get instant answers now!

16 y 8 1

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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 .)

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

Practice set c

Factor, if possible, the following trinomials.

m 2 - 8 m + 16

( m - 4 ) 2

Got questions? Get instant answers now!

k 2 + 10 k + 25

( k + 5 ) 2

Got questions? Get instant answers now!

4 a 2 + 12 a + 9

( 2 a + 3 ) 2

Got questions? Get instant answers now!

9 x 2 - 24 x y + 16 y 2

( 3 x - 4 y ) 2

Got questions? Get instant answers now!

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

2 w z ( w + 4 z ) 2

Got questions? Get instant answers now!

x 2 + 12 x + 49

not possible

Got questions? Get instant answers now!

Exercises

For the following problems, factor the binomials.

a 2 9

( a + 3 ) ( a 3 )

Got questions? Get instant answers now!

x 2 16

( x + 4 ) ( x 4 )

Got questions? Get instant answers now!

a 2 100

( a + 10 ) ( a 10 )

Got questions? Get instant answers now!

4 a 2 64

4 ( a + 4 ) ( a 4 )

Got questions? Get instant answers now!

3 x 2 27

3 ( x + 3 ) ( x 3 )

Got questions? Get instant answers now!

4 a 2 25

( 2 a + 5 ) ( 2 a 5 )

Got questions? Get instant answers now!

36 y 2 25

( 6 y + 5 ) ( 6 y 5 )

Got questions? Get instant answers now!

12 a 2 75

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

Got questions? Get instant answers now!

8 y 2 50

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

Got questions? Get instant answers now!

x 2 y 2 25

( x y + 5 ) ( x y 5 )

Got questions? Get instant answers now!

x 4 y 4 9 a 2

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

Got questions? Get instant answers now!

4 a 2 b 2 9 b 2

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

Got questions? Get instant answers now!

a 2 b 2

( a + b ) ( a b )

Got questions? Get instant answers now!

x 4 y 4

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

Got questions? Get instant answers now!

a 8 y 2

( a 4 + y ) ( a 4 y )

Got questions? Get instant answers now!

b 6 x 4

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

Got questions? Get instant answers now!

25 a 2

( 5 + a ) ( 5 a )

Got questions? Get instant answers now!

100 36 b 4

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

Got questions? Get instant answers now!

x 4 16

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

Got questions? Get instant answers now!

a 4 b 4

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

Got questions? Get instant answers now!

x 12 y 12

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

Got questions? Get instant answers now!

a 3 c 2 25 a c 2

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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 )

Got questions? Get instant answers now!

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

y 2 20 y 100

( y + 10 ) 2

Got questions? Get instant answers now!

x 2 10 x 25

( x 5 ) 2

Got questions? Get instant answers now!

a 2 24 a 144

( a 12 ) 2

Got questions? Get instant answers now!

9 x 2 6 x 1

( 3 x + 1 ) 2

Got questions? Get instant answers now!

16 a 2 24 a 9

( 4 a 3 ) 2

Got questions? Get instant answers now!

9 x 2 6 x y y 2

( 3 x + y ) 2

Got questions? Get instant answers now!

36 a 2 60 a b 25 b 2

( 6 a + 5 b ) 2

Got questions? Get instant answers now!

12 a 2 60 a 75

3 ( 2 a 5 ) 2

Got questions? Get instant answers now!

32 x 2 16 x 2

2 ( 4 x + 1 ) 2

Got questions? Get instant answers now!

4 a 2 a 9

not factorable

Got questions? Get instant answers now!

x 5 8 x 4 16 x 3

x 3 ( x + 4 ) 2

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

Exercises for review

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

( m 3 ) ( x y )

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

Questions & Answers

if three forces F1.f2 .f3 act at a point on a Cartesian plane in the daigram .....so if the question says write down the x and y components ..... I really don't understand
Syamthanda Reply
hey , can you please explain oxidation reaction & redox ?
Boitumelo Reply
hey , can you please explain oxidation reaction and redox ?
Boitumelo
for grade 12 or grade 11?
Sibulele
the value of V1 and V2
Tumelo Reply
advantages of electrons in a circuit
Rethabile Reply
we're do you find electromagnetism past papers
Ntombifuthi
what a normal force
Tholulwazi Reply
it is the force or component of the force that the surface exert on an object incontact with it and which acts perpendicular to the surface
Sihle
what is physics?
Petrus Reply
what is the half reaction of Potassium and chlorine
Anna Reply
how to calculate coefficient of static friction
Lisa Reply
how to calculate static friction
Lisa
How to calculate a current
Tumelo
how to calculate the magnitude of horizontal component of the applied force
Mogano
How to calculate force
Monambi
a structure of a thermocouple used to measure inner temperature
Anna Reply
a fixed gas of a mass is held at standard pressure temperature of 15 degrees Celsius .Calculate the temperature of the gas in Celsius if the pressure is changed to 2×10 to the power 4
Amahle Reply
How is energy being used in bonding?
Raymond Reply
what is acceleration
Syamthanda Reply
a rate of change in velocity of an object whith respect to time
Khuthadzo
how can we find the moment of torque of a circular object
Kidist
Acceleration is a rate of change in velocity.
Justice
t =r×f
Khuthadzo
how to calculate tension by substitution
Precious Reply
hi
Shongi
hi
Leago
use fnet method. how many obects are being calculated ?
Khuthadzo
khuthadzo hii
Hulisani
how to calculate acceleration and tension force
Lungile Reply
you use Fnet equals ma , newtoms second law formula
Masego
please help me with vectors in two dimensions
Mulaudzi Reply
how to calculate normal force
Mulaudzi
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elementary algebra' conversation and receive update notifications?

Ask