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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

He charges $125 per job. His monthly expenses are $1,600. How many jobs must he work in order to make a profit of at least $2,400?
Alicia Reply
at least 20
Ayla
what are the steps?
Alicia
6.4 jobs
Grahame
32
Grahame
what is algebra
Azhar Reply
repeated addition and subtraction of the order of operations. i love algebra I'm obsessed.
Shemiah
hi
Krekar
One-fourth of the candies in a bag of M&M’s are red. If there are 23 red candies, how many candies are in the bag?
Leanna Reply
rectangular field solutions
Navin Reply
What is this?
Donna
the proudact of 3x^3-5×^2+3 and 2x^2+5x-4 in z7[x]/ is
anas Reply
?
Choli
a rock is thrown directly upward with an initial velocity of 96feet per second from a cliff 190 feet above a beach. The hight of tha rock above the beach after t second is given by the equation h=_16t^2+96t+190
Usman
Stella bought a dinette set on sale for $725. The original price was $1,299. To the nearest tenth of a percent, what was the rate of discount?
Manhwa Reply
44.19%
Scott
40.22%
Terence
44.2%
Orlando
I don't know
Donna
if you want the discounted price subtract $725 from $1299. then divide the answer by $1299. you get 0.4419... but as percent you get 44.19... but to the nearest tenth... round .19 to .2 and you get 44.2%
Orlando
you could also just divide $725/$1299 and then subtract it from 1. then you get the same answer.
Orlando
p mulripied-5 and add 30 to it
Tausif Reply
p mulripied-5 and add30
Tausif
p mulripied-5 and addto30
Tausif
Can you explain further
Monica Reply
p mulripied-5 and add to 30
Tausif
-5p+30?
Corey
p=-5+30
Jacob
How do you find divisible numbers without a calculator?
Jacob Reply
TAKE OFF THE LAST DIGIT AND MULTIPLY IT 9. SUBTRACT IT THE DIGITS YOU HAVE LEFT. IF THE ANSWER DIVIDES BY 13(OR IS ZERO), THEN YOUR ORIGINAL NUMBER WILL ALSO DIVIDE BY 13!IS DIVISIBLE BY 13
BAINAMA
When she graduates college, Linda will owe $43,000 in student loans. The interest rate on the federal loans is 4.5% and the rate on the private bank loans is 2%. The total interest she owes for one year was $1,585. What is the amount of each loan?
Ariana Reply
Sean took the bus from Seattle to Boise, a distance of 506 miles. If the trip took 7 2/3 hours, what was the speed of the bus?
Kirisma Reply
66miles/hour
snigdha
How did you work it out?
Esther
s=mi/hr 2/3~0.67 s=506mi/7.67hr = ~66 mi/hr
Orlando
hello, I have algebra phobia. Subtracting negative numbers always seem to get me confused.
Alicia Reply
what do you need help in?
Felix
subtracting a negative....is adding!!
Heather
look at the numbers if they have different signs, it's like subtracting....but you keep the sign of the largest number...
Felix
for example.... -19 + 7.... different signs...subtract.... 12 keep the sign of the "largest" number 19 is bigger than 7.... 19 has the negative sign... Therefore, -12 is your answer...
Felix
—12
Niazmohammad
Thanks Felix.l also get confused with signs.
Esther
Thank you for this
Shatey
ty
Graham
think about it like you lost $19 (-19), then found $7(+7). Totally you lost just $12 (-12)
Annushka
I used to struggle a lot with negative numbers and math in general what I typically do is look at it in terms of money I have -$5 in my account I then take out 5 more dollars how much do I have in my account well-$10 ... I also for a long time would draw it out on a number line to visualize it
Meg
practicing with smaller numbers to understand then working with larger numbers helps too and the song/rhyme same sign add and keep opposite signs subtract keep the sign of the bigger # then you'll be exact
Meg
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
John Reply
168.50=R
Heather
john is 5years older than wanjiru.the sum of their years is27years.what is the age of each
achol Reply
46
mustee
j 17 w 11
Joseph
john is 16. wanjiru is 11.
Felix
27-5=22 22÷2=11 11+5=16
Joyce
where's the answers?
Ed Reply
I don't see where the answers are.
Ed
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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