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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 did you get 1640
Noor Reply
If auger is pair are the roots of equation x2+5x-3=0
Peter Reply
Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.
MATTHEW Reply
420
Sharon
from theory: distance [miles] = speed [mph] × time [hours] info #1 speed_Dennis × 1.5 = speed_Wayne × 2 => speed_Wayne = 0.75 × speed_Dennis (i) info #2 speed_Dennis = speed_Wayne + 7 [mph] (ii) use (i) in (ii) => [...] speed_Dennis = 28 mph speed_Wayne = 21 mph
George
Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5. Substituting the first equation into the second: W * 2 = (W + 7) * 1.5 W * 2 = W * 1.5 + 7 * 1.5 0.5 * W = 7 * 1.5 W = 7 * 3 or 21 W is 21 D = W + 7 D = 21 + 7 D = 28
Salma
Devon is 32 32​​ years older than his son, Milan. The sum of both their ages is 54 54​. Using the variables d d​ and m m​ to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.
Aaron Reply
find product (-6m+6) ( 3m²+4m-3)
SIMRAN Reply
-42m²+60m-18
Salma
what is the solution
bill
how did you arrive at this answer?
bill
-24m+3+3mÁ^2
Susan
i really want to learn
Amira
I only got 42 the rest i don't know how to solve it. Please i need help from anyone to help me improve my solving mathematics please
Amira
Hw did u arrive to this answer.
Aphelele
hi
Bajemah
-6m(3mA²+4m-3)+6(3mA²+4m-3) =-18m²A²-24m²+18m+18mA²+24m-18 Rearrange like items -18m²A²-24m²+42m+18A²-18
Salma
complete the table of valuesfor each given equatio then graph. 1.x+2y=3
Jovelyn Reply
x=3-2y
Salma
y=x+3/2
Salma
Hi
Enock
given that (7x-5):(2+4x)=8:7find the value of x
Nandala
3x-12y=18
Kelvin
please why isn't that the 0is in ten thousand place
Grace Reply
please why is it that the 0is in the place of ten thousand
Grace
Send the example to me here and let me see
Stephen
A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg
Marry Reply
how far
Abubakar
cool u
Enock
state in which quadrant or on which axis each of the following angles given measure. in standard position would lie 89°
Abegail Reply
hello
BenJay
hi
Method
I am eliacin, I need your help in maths
Rood
how can I help
Sir
hmm can we speak here?
Amoon
however, may I ask you some questions about Algarba?
Amoon
hi
Enock
what the last part of the problem mean?
Roger
The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.
cameron Reply
Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.
mahnoor Reply
I'm guessing, but it's somewhere around $4335.00 I think
Lewis
12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67. Check: Sales = 3542 Commission 12%=425.04 Pay = 500 + 425.04 = 925.04. 925.04 > 925.00
Munster
difference between rational and irrational numbers
Arundhati Reply
When traveling to Great Britain, Bethany exchanged $602 US dollars into £515 British pounds. How many pounds did she receive for each US dollar?
Jakoiya Reply
how to reduced echelon form
Solomon Reply
Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?
Zack Reply
d=r×t the equation would be 8/r+24/r+4=3 worked out
Sheirtina
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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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