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Factor: −16 z 64 .

−8 ( 8 z + 8 )

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Factor: −9 y 27 .

−9 ( y + 3 )

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Factor: −6 a 2 + 36 a .

Solution

The leading coefficient is negative, so the GCF will be negative.?

Since the leading coefficient is negative, the GCF is negative, −6 a .

.
.

Rewrite each term using the GCF. .
Factor the GCF. .
Check.
−6 a ( a 6 )
−6 a a + ( −6 a ) ( −6 )
−6 a 2 + 36 a

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Factor: −4 b 2 + 16 b .

−4 b ( b 4 )

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Factor: −7 a 2 + 21 a .

−7 a ( a 3 )

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Factor: 5 q ( q + 7 ) 6 ( q + 7 ) .

Solution

The GCF is the binomial q + 7 .

.
Factor the GCF, ( q + 7). .
Check on your own by multiplying.

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Factor: 4 m ( m + 3 ) 7 ( m + 3 ) .

( m + 3 ) ( 4 m 7 )

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Factor: 8 n ( n 4 ) + 5 ( n 4 ) .

( n 4 ) ( 8 n + 5 )

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Factor by grouping

When there is no common factor of all the terms of a polynomial, look for a common factor in just some of the terms. When there are four terms, a good way to start is by separating the polynomial into two parts with two terms in each part. Then look for the GCF in each part. If the polynomial can be factored, you will find a common factor emerges from both parts.

(Not all polynomials can be factored. Just like some numbers are prime, some polynomials are prime.)

How to factor by grouping

Factor: x y + 3 y + 2 x + 6 .

Solution

This table gives the steps for factoring x y + 3 y + 2 x + 6. In the first row there is the statement, “group terms with common factors”. In the next column, there is the statement of no common factors of all 4 terms. The last column shows the first two terms grouped and the last two terms grouped. The second row has the statement, “factor out the common factor from each group”. The second column in the second row states to factor out the GCF from the two separate groups. The third column in the second row has the expression y(x + 3) + 2(x + 3). The third row has the statement, “factor the common factor from the expression”. The second column in this row points out there is a common factor of (x + 3). The third column in the third row shows the factor of (x + 3) factored from the two groups, (x + 3) times (y + 2). The last row has the statement, “check”. The second column in this row states to multiply (x + 3)(y + 2). The product is shown in the last column of the original polynomial x y + 3 y + 2 x + 6.
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Factor: x y + 8 y + 3 x + 24 .

( x + 8 ) ( y + 3 )

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Factor: a b + 7 b + 8 a + 56 .

( a + 7 ) ( b + 8 )

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Factor by grouping.

  1. Group terms with common factors.
  2. Factor out the common factor in each group.
  3. Factor the common factor from the expression.
  4. Check by multiplying the factors.

Factor: x 2 + 3 x 2 x 6 .

Solution

There is no GCF in all four terms. x 2 + 3 x −2 x 6 Separate into two parts. x 2 + 3 x −2 x 6 Factor the GCF from both parts. Be careful with the signs when factoring the GCF from the last two terms. x ( x + 3 ) 2 ( x + 3 ) ( x + 3 ) ( x 2 ) Check on your own by multiplying.

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Factor: x 2 + 2 x 5 x 10 .

( x 5 ) ( x + 2 )

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Factor: y 2 + 4 y 7 y 28 .

( y + 4 ) ( y 7 )

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Access these online resources for additional instruction and practice with greatest common factors (GFCs) and factoring by grouping.

Key concepts

  • Finding the Greatest Common Factor (GCF): To find the GCF of two expressions:
    1. Factor each coefficient into primes. Write all variables with exponents in expanded form.
    2. List all factors—matching common factors in a column. In each column, circle the common factors.
    3. Bring down the common factors that all expressions share.
    4. Multiply the factors as in [link] .
  • Factor the Greatest Common Factor from a Polynomial: To factor a greatest common factor from a polynomial:
    1. Find the GCF of all the terms of the polynomial.
    2. Rewrite each term as a product using the GCF.
    3. Use the ‘reverse’ Distributive Property to factor the expression.
    4. Check by multiplying the factors as in [link] .
  • Factor by Grouping: To factor a polynomial with 4 four or more terms
    1. Group terms with common factors.
    2. Factor out the common factor in each group.
    3. Factor the common factor from the expression.
    4. Check by multiplying the factors as in [link] .

Practice makes perfect

Find the Greatest Common Factor of Two or More Expressions

In the following exercises, find the greatest common factor.

10 p 3 q , 12 p q 2

2 p q

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12 m 2 n 3 , 30 m 5 n 3

6 m 2 n 3

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10 a 3 , 12 a 2 , 14 a

2 a

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35 x 3 , 10 x 4 , 5 x 5

5 x 3

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Factor the Greatest Common Factor from a Polynomial

In the following exercises, factor the greatest common factor from each polynomial.

9 n 63

9 ( n 7 )

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3 x 2 + 6 x 9

3 ( x 2 + 2 x 3 )

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8 p 2 + 4 p + 2

2 ( 4 p 2 + 2 p + 1 )

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8 y 3 + 16 y 2

8 y 2 ( y + 2 )

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5 x 3 15 x 2 + 20 x

5 x ( x 2 3 x + 4 )

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12 x y 2 + 18 x 2 y 2 30 y 3

6 y 2 ( 2 x + 3 x 2 5 y )

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21 p q 2 + 35 p 2 q 2 28 q 3

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−2 x 4

−2 ( x + 4 )

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5 x ( x + 1 ) + 3 ( x + 1 )

( x + 1 ) ( 5 x + 3 )

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2 x ( x 1 ) + 9 ( x 1 )

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3 b ( b 2 ) 13 ( b 2 )

( b 2 ) ( 3 b 13 )

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6 m ( m 5 ) 7 ( m 5 )

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Factor by Grouping

In the following exercises, factor by grouping.

x y + 2 y + 3 x + 6

( y + 3 ) ( x + 2 )

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u v 9 u + 2 v 18

( u + 2 ) ( v 9 )

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p q 10 p + 8 q 80

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b 2 + 5 b 4 b 20

( b 4 ) ( b + 5 )

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m 2 + 6 m 12 m 72

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p 2 + 4 p 9 p 36

( p 9 ) ( p + 4 )

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x 2 + 5 x 3 x 15

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Mixed Practice

In the following exercises, factor.

−20 x 10

−10 ( 2 x + 1 )

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3 x 3 7 x 2 + 6 x 14

( x 2 + 2 ) ( 3 x 7 )

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x 2 + x y + 5 x + 5 y

( x + y ) ( x + 5 )

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5 x 3 3 x 2 5 x 3

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Everyday math

Area of a rectangle The area of a rectangle with length 6 less than the width is given by the expression w 2 6 w , where w = width. Factor the greatest common factor from the polynomial.

w ( w 6 )

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Height of a baseball The height of a baseball t seconds after it is hit is given by the expression −16 t 2 + 80 t + 4 . Factor the greatest common factor from the polynomial.

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Writing exercises

The greatest common factor of 36 and 60 is 12. Explain what this means.

Answers will vary.

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What is the GCF of y 4 , y 5 , and y 10 ? Write a general rule that tells you how to find the GCF of y a , y b , and y c .

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Self check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This table has the following statements all to be preceded by “I can…”. The first is “find the greatest common factor of two or more expressions”. The second is “factor the greatest common factor from a polynomial”. The third is “factor by grouping”. In the columns beside these statements are the headers, “confidently”, “with some help”, and “no-I don’t get it!”.

If most of your checks were:

…confidently. Congratulations! You have achieved your goals in this section! Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific!

…with some help. This must be addressed quickly as topics you do not master become potholes in your road to success. Math is sequential—every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no - I don’t get it! This is critical and you must not ignore it. You need to get help immediately or you will quickly be overwhelmed. See your instructor as soon as possible to discuss your situation. Together you can come up with a plan to get you the help you need.

Questions & Answers

Bruce drives his car for his job. The equation R=0.575m+42 models the relation between the amount in dollars, R, that he is reimbursed and the number of miles, m, he drives in one day. Find the amount Bruce is reimbursed on a day when he drives 220 miles.
Dojzae Reply
LeBron needs 150 milliliters of a 30% solution of sulfuric acid for a lab experiment but only has access to a 25% and a 50% solution. How much of the 25% and how much of the 50% solution should he mix to make the 30% solution?
Xona Reply
5%
Michael
hey everyone how to do algebra
The Reply
Felecia answer 1.5 hours before he reaches her
Adriana Reply
I would like to solve the problem -6/2x
rachel Reply
12x
Andrew
how
Christian
Does the x represent a number or does it need to be graphed ?
latonya
-3/x
Venugopal
-3x is correct
Atul
Arnold invested $64,000, some at 5.5% interest and the rest at 9%. How much did he invest at each rate if he received $4,500 in interest in one year?
Stephanie Reply
Tickets for the community fair cost $12 for adults and $5 for children. On the first day of the fair, 312 tickets were sold for a total of $2204. How many adult tickets and how many child tickets were sold?
Alpha Reply
220
gayla
Three-fourths of the people at a concert are children. If there are 87 children, what is the total number of people at the concert?
Tsimmuaj Reply
Erica earned a total of $50,450 last year from her two jobs. The amount she earned from her job at the store was $1,250 more than four times the amount she earned from her job at the college. How much did she earn from her job at the college?
Tsimmuaj
Erica earned a total of $50,450 last year from her two jobs. The amount she earned from her job at the store was $1,250 more than four times the amount she earned from her job at the college. How much did she earn from her job at the college?
Tsimmuaj
? Is there anything wrong with this passage I found the total sum for 2 jobs, but found why elaborate on extra If I total one week from the store *4 would = the month than the total is = x than x can't calculate 10 month of a year
candido
what would be wong
candido
87 divided by 3 then multiply that by 4. 116 people total.
Melissa
the actual number that has 3 out of 4 of a whole pie
candido
was having a hard time finding
Teddy
use Matrices for the 2nd question
Daniel
One number is 11 less than the other number. If their sum is increased by 8, the result is 71. Find the numbers.
Tsimmuaj Reply
26 + 37 = 63 + 8 = 71
gayla
26+37=63+8=71
ziad
11+52=63+8=71
Thisha
how do we know the answer is correct?
Thisha
23 is 11 less than 37. 23+37=63. 63+8=71. that is what the question asked for.
gayla
23 +11 = 37. 23+37=63 63+8=71
Gayla
by following the question. one number is 11 less than the other number 26+11=37 so 26+37=63+8=71
Gayla
your answer did not fit the guidelines of the question 11 is 41 less than 52.
gayla
71-8-11 =52 is this correct?
Ruel
let the number is 'x' and the other number is "x-11". if their sum is increased means: x+(x-11)+8 result will be 71. so x+(x-11)+8=71 2x-11+8=71 2x-3=71 2x=71+3 2x=74 1/2(2x=74)1/2 x=37 final answer
tesfu
just new
Muwanga
Amara currently sells televisions for company A at a salary of $17,000 plus a $100 commission for each television she sells. Company B offers her a position with a salary of $29,000 plus a $20 commission for each television she sells. How televisions would Amara need to sell for the options to be equal?
Tsimmuaj Reply
yes math
Kenneth
company A 13 company b 5. A 17,000+13×100=29,100 B 29,000+5×20=29,100
gayla
need help with math to do tsi test
Toocute
me too
Christian
have you tried the TSI practice test ***tsipracticetest.com
gayla
DaMarcus and Fabian live 23 miles apart and play soccer at a park between their homes. DaMarcus rode his bike for 34 of an hour and Fabian rode his bike for 12 of an hour to get to the park. Fabian’s speed was 6 miles per hour faster than DaMarcus’s speed. Find the speed of both soccer players.
gustavo Reply
?
Ann
DaMarcus: 16 mi/hr Fabian: 22 mi/hr
Sherman
Joy is preparing 20 liters of a 25% saline solution. She has only a 40% solution and a 10% solution in her lab. How many liters of the 40% solution and how many liters of the 10% solution should she mix to make the 25% solution?
Wenda Reply
15 and 5
32 is 40% , & 8 is 10 % , & any 4 letters is 5%.
Karen
It felt that something is missing on the question like: 40% of what solution? 10% of what solution?
Jhea
its confusing
Sparcast
3% & 2% to complete the 25%
Sparcast
because she already has 20 liters.
Sparcast
ok I was a little confused I agree 15% & 5%
Sparcast
8,2
Karen
Jim and Debbie earned $7200. Debbie earned $1600 more than Jim earned. How much did they earned
Arleathia Reply
5600
Gloria
1600
Gloria
Bebbie: 4,400 Jim: 2,800
Jhea
A river cruise boat sailed 80 miles down the Mississippi River for 4 hours. It took 5 hours to return. Find the rate of the cruise boat in still water and the rate of the current.
Sunnyshay Reply
A veterinarian is enclosing a rectangular outdoor running area against his building for the dogs he cares for. He needs to maximize the area using 100 feet of fencing. The quadratic equation A=x(100−2x) gives the area, A , of the dog run for the length, x , of the building that will border the dog run. Find the length of the building that should border the dog run to give the maximum area, and then find the maximum area of the dog run.
Elizabeth Reply
ggfcc
Mike
Practice Key Terms 2

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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