# 7.1 Greatest common factor and factor by grouping

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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 $-3\left(6a+11\right)$ .
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    . 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    Notice that, because the GCF is a factor of both numbers, 54 and 36 can be written as multiples of 18.

$\begin{array}{c}54=18·3\hfill \\ 36=18·2\hfill \end{array}$

Find the GCF of 48 and 80.

16

Find the GCF of 18 and 40.

2

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 $27{x}^{3}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}18{x}^{4}$ .

## 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 $9{x}^{3}.$

Find the GCF: $12{x}^{2},18{x}^{3}$ .

$3{x}^{2}$

Find the GCF: $16{y}^{2},24{y}^{3}$ .

$8{y}^{2}$

Find the GCF of $4{x}^{2}y,6x{y}^{3}$ .

## 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 $6x{y}^{3}$ is $2\mathrm{xy}$ .

Find the GCF: $6a{b}^{4},8{a}^{2}b$ .

$2ab$

Find the GCF: $9{m}^{5}{n}^{2},12{m}^{3}n$ .

$3{m}^{3}n$

Find the GCF of: $21{x}^{3},9{x}^{2},15x$ .

## 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 $15x$ is $3x.$

Find the greatest common factor: $25{m}^{4},35{m}^{3},20{m}^{2}$ .

$5{m}^{2}$

Find the greatest common factor: $14{x}^{3},70{x}^{2},105x$ .

$7x$

## 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 $2·6\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}3·4\right),$ 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:

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?
rectangular field solutions
What is this?
Donna
the proudact of 3x^3-5×^2+3 and 2x^2+5x-4 in z7[x]/ is
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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?
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
168.50=R
Heather
john is 5years older than wanjiru.the sum of their years is27years.what is the age of each
46
mustee
j 17 w 11
Joseph
john is 16. wanjiru is 11.
Felix
27-5=22 22÷2=11 11+5=16
Joyce
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?
3
Christopher
18t+14t=96 32t=96 32/96 3
Christopher
show that a^n-b^2n is divisible by a-b By By By OpenStax By OpenStax By Sam Luong By Rhodes By OpenStax By Mary Cohen By Brooke Delaney By Melinda Salzer