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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

Geneva treated her parents to dinner at their favorite restaurant. The bill was $74.25. Geneva wants to leave 16 % of the total - bill as a tip. How much should the tip be?
74.25 × .16 then get the total and that will be your tip
David
$74.25 x 0.16 = $11.88 total bill: $74.25 + $11.88 = $86.13
ericka
yes and tip 16% will be $11.88
David
what is the shorter way to do it
Cesar Reply
Priam has dimes and pennies in a cup holder in his car. The total value of the coins is $4.21. The number of dimes is three less than four times the number of pennies. How many dimes and how many pennies are in the cup?
Alexandra Reply
Uno de los ángulos suplementario es 4° más que 1/3 del otro ángulo encuentra las medidas de cada uno de los angulos
Enith Reply
June needs 48 gallons of punch for a party and has two different coolers to carry it in. The bigger cooler is five times as large as the smaller cooler. How many gallons can each cooler hold?
Alexandra Reply
I hope this is correct, x=cooler 1 5x=cooler 2 x + 5x = 48 6x=48 ×=8 gallons 5×=40 gallons
ericka
Priam has pennies and dimes in a cup holder in his car. The total value of the coins is $4.21 . The number of dimes is three less than four times the number of pennies. How many pennies and how many dimes are in the cup?
Cecilia Reply
Arnold invested $64,000 some at 5.5% interest and the rest at 9% interest how much did he invest at each rate if he received $4500 in interest in one year
Heidi Reply
List five positive thoughts you can say to yourself that will help youapproachwordproblemswith a positive attitude. You may want to copy them on a sheet of paper and put it in the front of your notebook, where you can read them often.
Elbert Reply
Avery and Caden have saved $27,000 towards a down payment on a house. They want to keep some of the money in a bank account that pays 2.4% annual interest and the rest in a stock fund that pays 7.2% annual interest. How much should they put into each account so that they earn 6% interest per year?
Leika Reply
324.00
Irene
1.2% of 27.000
Irene
i did 2.4%-7.2% i got 1.2%
Irene
I have 6% of 27000 = 1620 so we need to solve 2.4x +7.2y =1620
Catherine
I think Catherine is on the right track. Solve for x and y.
Scott
next bit : x=(1620-7.2y)/2.4 y=(1620-2.4x)/7.2 I think we can then put the expression on the right hand side of the "x=" into the second equation. 2.4x in the second equation can be rewritten as 2.4(rhs of first equation) I write this out tidy and get back to you...
Catherine
Darrin is hanging 200 feet of Christmas garland on the three sides of fencing that enclose his rectangular front yard. The length is five feet less than five times the width. Find the length and width of the fencing.
Ericka Reply
Mario invested $475 in $45 and $25 stock shares. The number of $25 shares was five less than three times the number of $45 shares. How many of each type of share did he buy?
Jawad Reply
let # of $25 shares be (x) and # of $45 shares be (y) we start with $25x + $45y=475, right? we are told the number of $25 shares is 3y-5) so plug in this for x. $25(3y-5)+$45y=$475 75y-125+45y=475 75y+45y=600 120y=600 y=5 so if #$25 shares is (3y-5) plug in y.
Joshua
will every polynomial have finite number of multiples?
cricket Reply
a=# of 10's. b=# of 20's; a+b=54; 10a + 20b=$910; a=54 -b; 10(54-b) + 20b=$910; 540-10b+20b=$910; 540+10b=$910; 10b=910-540; 10b=370; b=37; so there are 37 20's and since a+b=54, a+37=54; a=54-37=17; a=17, so 17 10's. So lets check. $740+$170=$910.
David Reply
. A cashier has 54 bills, all of which are $10 or $20 bills. The total value of the money is $910. How many of each type of bill does the cashier have?
jojo Reply
whats the coefficient of 17x
Dwayne Reply
the solution says it 14 but how i thought it would be 17 im i right or wrong is the exercise wrong
Dwayne
17
Melissa
wow the exercise told me 17x solution is 14x lmao
Dwayne
thank you
Dwayne
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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