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By the end of this section, you will be able to:
  • Find the greatest common factor of two or more expressions
  • Factor the greatest common factor from a polynomial
  • Factor by grouping

Before you get started, take this readiness quiz.

  1. Factor 56 into primes.
    If you missed this problem, review [link] .
  2. Find the least common multiple of 18 and 24.
    If you missed this problem, review [link] .
  3. Simplify −3 ( 6 a + 11 ) .
    If you missed this problem, review [link] .

Find the greatest common factor of two or more expressions

Earlier we multiplied factors together to get a product. Now, we will be reversing this process; we will start with a product and then break it down into its factors. Splitting a product into factors is called factoring    .

This figure has two factors being multiplied. They are 8 and 7. Beside this equation there are other factors multiplied. They are 2x and (x+3). The product is given as 2x^2 plus 6x. Above the figure is an arrow towards the right with multiply inside. Below the figure is an arrow to the left with factor inside.

We have learned how to factor numbers to find the least common multiple (LCM) of two or more numbers. Now we will factor expressions and find the greatest common factor    of two or more expressions. The method we use is similar to what we used to find the LCM.

Greatest common factor

The greatest common factor    (GCF) of two or more expressions is the largest expression that is a factor of all the expressions.

First we’ll find the GCF of two numbers.

How to find the greatest common factor of two or more expressions

Find the GCF of 54 and 36.

Solution

This table has three columns. In the first column are the steps for factoring. The first row has the first step, factor each coefficient into primes and write all variables with exponents in expanded form. The second column in the first row has “factor 54 and 36”. The third column in the first row has 54 and 36 factored with factor trees. The prime factors of 54 are circled and are 3, 3, 2, and3. The prime factors of 36 are circled and are 2,3,2,3. The second row has the second step of “in each column, circle the common factors. The second column in the second row has the statement “circle the 2, 3 and 3 that are shared by both numbers”. The third column in the second row has the prime factors of 36 and 54 in rows above each other. The common factors of 2, 3, and 3 are circled. The third row has the step “bring down the common factors that all expressions share”. The second column in the third row has “bring down the 2,3, and 3 then multiply”. The third column in the third row has “GCF = 2 times 3 times 3”. The fourth row has the fourth step “multiply the factors”. The second column in the fourth row is blank. The third column in the fourth row has “GCF = 18” and “the GCF of 54 and 36 is 18”.

Notice that, because the GCF is a factor of both numbers, 54 and 36 can be written as multiples of 18.

54 = 18 · 3 36 = 18 · 2
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Find the GCF of 48 and 80.

16

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Find the GCF of 18 and 40.

2

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We summarize the steps we use to find the GCF below.

Find the greatest common factor (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.

In the first example, the GCF was a constant. In the next two examples, we will get variables in the greatest common factor.

Find the greatest common factor of 27 x 3 and 18 x 4 .

Solution

Factor each coefficient into primes and write the variables with exponents in expanded form. Circle the common factors in each column. .
Bring down the common factors. .
Multiply the factors. .
The GCF of 27 x 3 and 18 x 4 is 9 x 3 .

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Find the GCF: 12 x 2 , 18 x 3 .

3 x 2

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Find the GCF: 16 y 2 , 24 y 3 .

8 y 2

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Find the GCF of 4 x 2 y , 6 x y 3 .

Solution

Factor each coefficient into primes and write the variables with exponents in expanded form. Circle the common factors in each column. .
Bring down the common factors. .
Multiply the factors. .
The GCF of 4 x 2 y and 6 x y 3 is 2 xy .

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Find the GCF: 6 a b 4 , 8 a 2 b .

2 a b

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Find the GCF: 9 m 5 n 2 , 12 m 3 n .

3 m 3 n

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Find the GCF of: 21 x 3 , 9 x 2 , 15 x .

Solution

Factor each coefficient into primes and write the variables with exponents in expanded form. Circle the common factors in each column. .
Bring down the common factors. .
Multiply the factors. .
The GCF of 21 x 3 , 9 x 2 and 15 x is 3 x .

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Find the greatest common factor: 25 m 4 , 35 m 3 , 20 m 2 .

5 m 2

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Find the greatest common factor: 14 x 3 , 70 x 2 , 105 x .

7 x

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Factor the greatest common factor from a polynomial

Just like in arithmetic, where it is sometimes useful to represent a number in factored form (for example, 12 as 2 · 6 or 3 · 4 ) , in algebra, it can be useful to represent a polynomial in factored form. One way to do this is by finding the GCF of all the terms. Remember, we multiply a polynomial by a monomial as follows:

Questions & Answers

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
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
Cindy and Richard leave their dorm in Charleston at the same time. Cindy rides her bicycle north at a speed of 18 miles per hour. Richard rides his bicycle south at a speed of 14 miles per hour. How long will it take them to be 96 miles apart?
Maddy Reply
3
Christopher
18t+14t=96 32t=96 32/96 3
Christopher
show that a^n-b^2n is divisible by a-b
Florence Reply
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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