# 6.2: Greatest Common Factor and Factor by Grouping - Mathematics

Learning Objectives

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 (LCM) of 18 and 24.
If you missed this problem, review [link].
3. Multiply: (−3a(7a+8b)).
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 reverse 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.

We summarize the steps we use to find the greatest common factor.

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.

The next example will show us the steps to find the greatest common factor of three expressions.

Example (PageIndex{1})

Find the greatest common factor of (21x^3,space 9x^2,space 15x).

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

Example (PageIndex{2})

Find the greatest common factor: (25m^4,space 35m^3,space 20m^2.)

(5m^2)

Example (PageIndex{3})

Find the greatest common factor: (14x^3,space 70x^2,space 105x).

(7x)

## Factor the Greatest Common Factor from a Polynomial

It is sometimes useful to represent a number as a product of factors, for example, 12 as (2·6) or (3·4). In algebra, it can also be useful to represent a polynomial in factored form. We will start with a product, such as (3x^2+15x), and end with its factors, (3x(x+5)). To do this we apply the Distributive Property “in reverse.”

We state the Distributive Property here just as you saw it in earlier chapters and “in reverse.”

DISTRIBUTIVE PROPERTY

If a, b, and c are real numbers, then

The form on the left is used to multiply. The form on the right is used to factor.

So how do you use the Distributive Property to factor a polynomial? You just find the GCF of all the terms and write the polynomial as a product!

Example (PageIndex{4}): How to Use the Distributive Property to factor a polynomial

Factor: (8m^3−12m^2n+20mn^2).    Example (PageIndex{5})

Factor: (9xy^2+6x^2y^2+21y^3).

(3y^2(3x+2x^2+7y))

Example (PageIndex{6})

Factor: (3p^3−6p^2q+9pq^3).

(3p(p^2−2pq+3q^2))

FACTOR THE 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.

FACTOR AS A NOUN AND A VERB

We use “factor” as both a noun and a verb:

[egin{array} {ll} ext{Noun:} &hspace{50mm} 7 ext{ is a factor of }14 ext{Verb:} &hspace{50mm} ext{factor }3 ext{ from }3a+3end{array} onumber]

Example (PageIndex{7})

Factor: (5x^3−25x^2).

 Find the GCF of (5x^3) and (25x^2).   Rewrite each term. Factor the GCF. Check:[5x^2(x−5) onumber][5x^2·x−5x^2·5 onumber][5x^3−25x^2 checkmark onumber]

Example (PageIndex{8})

Factor: (2x^3+12x^2).

(2x^2(x+6))

Example (PageIndex{9})

Factor: (6y^3−15y^2).

(3y^2(2y−5))

Example (PageIndex{10})

Factor: (8x^3y−10x^2y^2+12xy^3).

 The GCF of (8x^3y,space −10x^2y^2,) and (12xy^3)is (2xy).   Rewrite each term using the GCF, (2xy). Factor the GCF. Check:[2xy(4x^2−5xy+6y^2) onumber][2xy·4x^2−2xy·5xy+2xy·6y^2 onumber][8x^3y−10x^2y^2+12xy^3checkmark onumber]

Example (PageIndex{11})

Factor: (15x^3y−3x^2y^2+6xy^3).

(3xy(5x^2−xy+2y^2))

Example (PageIndex{12})

Factor: (8a^3b+2a^2b^2−6ab^3).

(2ab(4a^2+ab−3b^2))

When the leading coefficient is negative, we factor the negative out as part of the GCF.

Example (PageIndex{13})

Factor: (−4a^3+36a^2−8a).

The leading coefficient is negative, so the GCF will be negative. Rewrite each term using the GCF, (−4a). Factor the GCF. Check: [−4a(a^2−9a+2) onumber][−4a·a^2−(−4a)·9a+(−4a)·2 onumber][−4a^3+36a^2−8acheckmark onumber]

Example (PageIndex{14})

Factor: (−4b^3+16b^2−8b).

(−4b(b^2−4b+2))

Example (PageIndex{15})

Factor: (−7a^3+21a^2−14a).

(−7a(a^2−3a+2))

So far our greatest common factors have been monomials. In the next example, the greatest common factor is a binomial.

Example (PageIndex{16})

Factor: (3y(y+7)−4(y+7)).

The GCF is the binomial (y+7). Factor the GCF, ((y+7)). Check on your own by multiplying.

Example (PageIndex{17})

Factor: (4m(m+3)−7(m+3)).

((m+3)(4m−7))

Example (PageIndex{18})

Factor: (8n(n−4)+5(n−4)).

((n−4)(8n+5))

## Factor by Grouping

Sometimes there is no common factor of all the terms of a polynomial. When there are four terms we separate 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.

Example (PageIndex{19}): How to Factor a Polynomial by Grouping

Factor by grouping: (xy+3y+2x+6).    Example (PageIndex{20})

Factor by grouping: (xy+8y+3x+24).

((x+8)(y+3))

Example (PageIndex{21})

Factor by grouping: (ab+7b+8a+56).

((a+7)(b+8))

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.

Example (PageIndex{22})

Factor by grouping: ⓐ (x^2+3x−2x−6) ⓑ (6x^2−3x−4x+2).

(egin{array} {ll} ext{There is no GCF in all four terms.} &x^2+3x−2x−6 ext{Separate into two parts.} &x^2+3xquad −2x−6 egin{array} {l} ext{Factor the GCF from both parts. Be careful} ext{with the signs when factoring the GCF from} ext{the last two terms.} end{array} &x(x+3)−2(x+3) ext{Factor out the common factor.} &(x+3)(x−2) ext{Check on your own by multiplying.} & end{array})

(egin{array} {ll} ext{There is no GCF in all four terms.} &6x^2−3x−4x+2 ext{Separate into two parts.} &6x^2−3xquad −4x+2 ext{Factor the GCF from both parts.} &3x(2x−1)−2(2x−1) ext{Factor out the common factor.} &(2x−1)(3x−2) ext{Check on your own by multiplying.} & end{array})

Example (PageIndex{23})

Factor by grouping: ⓐ (x^2+2x−5x−10) ⓑ (20x^2−16x−15x+12).

ⓐ ((x−5)(x+2))
ⓑ ((5x−4)(4x−3))

Example (PageIndex{24})

Factor by grouping: ⓐ (y^2+4y−7y−28) ⓑ (42m^2−18m−35m+15).

ⓐ ((y+4)(y−7))
ⓑ ((7m−3)(6m−5))

## Key Concepts

• How to find the greatest common factor (GCF) of two expressions.
1. Factor each coefficient into primes. In each column, circle the common factors.
2. Bring down the common factors that all expressions share.
3. Multiply the factors.
• Distributive Property: If (a), (b) and (c) are real numbers, then

The form on the left is used to multiply. The form on the right is used to factor.
• How to factor the 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.
• Factor as a Noun and a Verb: We use “factor” as both a noun and a verb.

[egin{array} {ll} ext{Noun:} &quad 7 ext{ is a factor of } 14 ext{Verb:} &quad ext{factor }3 ext{ from }3a+3end{array} onumber]

• How to 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.

### Glossary

factoring
Splitting a product into factors is called factoring.
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.

## Factoring

I. Greatest Common Factor. Always check to see if you can factor out the greatest common factor (GCF).
The greatest common factor is the largest factor that is shared by all the terms in the given expression. The
GCF may include variables. Also, the GCF sometimes contains more than one term.

 The GCF is 5x 2 . The GCF is (x – y).

After you determine the GCF, you may use the distributive property to rewrite the expression with the GCF
factored out.

 Recognize 4w as the GCF. Use distributive property to write in factored form.

Now we will consider three types of polynomials: binomial expressions (two terms), trinomial expressions
(three terms), expressions with four terms. The first step for all these cases will be to factor out the GCF.

II. Binomials. There are three special cases that fall under the two-term category.

A. Difference of squares. A 2 – B 2 = (A + B) (A – B)
This may be verified by multiplying out
the right hand side.

 This step may help you to see what the bases are Use the formula to rewrite in factored form.

B. Difference of cubes. A 3 – B 3 = (A – B) (A 2 + AB + B 2 )
Again, this may be verified by
multiplying out the right hand side.

 Factor out the GCF first. Recognize the difference of cubes. Write in factored form using the difference of cubes formula.

C. Sum of cubes. A 3 + B 3 = (A + B) (A2 – AB + B 2 )

 Recognize the sum of cubes. Write in factored form using the sum of cubes formula.

Note: There is no factorization for the sum of squares. For example, 9p 2 + 4q 2 cannot be factored. It is
prime.

III. Trinomials. We will discuss two different ways to factor a trinomial of the form ax 2 +bx+c.

A. The ac method or grouping method.
1. This is sometimes called the ac method because with trinomials of the form ax 2 +bx+c (where a,
b, c are constants) the first step will be to multiply a and c.
2. Next, you will look for two factors of the product “ac” that add to form the middle term’s
coefficient, “b” of the original trinomial.
3. Then you rewrite the middle term as the sum of those two factors you discovered in step 2. Don’t
forget to include the variable (they are like terms and need to be like the original middle term).
4. Now you have a four-term polynomial. Group the expression into two groups of two terms each
and factor out the GCF for each 2-term group. (This grouping step is the reason why we
sometimes call this the grouping method.)
5. You should now recognize a common binomial factor. Factor this binomial out and write the
expression in factored form by using the distributive property.

 First factor out the GCF. Then multiply “a” and “c.” (4)(-3) = -12 Two factors of -12 that add to form 4 are -2 and 6. Rewrite the middle term as the sum of “𔃀x” and 𔄞x.” Group the four terms into two groups of two. Factor out the GCF for each group and recognize (2x – 1) is the common binomial factor. Use the distributive property to write in factored form.

B. Trial and error method. This method involves finding factors of the leading term (the “a”) and the last
term (the “c”) and trying them out in the product of two binomials. Use FOIL to multiply and see if the
factors in your trial produce the original trinomial.

 Factors of 2 are 1&2. Factors of -12 are 1&-12, -1&12, 2&-6, -2&6, 3&-4, -3&4. Trying 1&-12. FOIL shows that this trial doesn’t work Trying 4&-3. FOIL shows this doesn’t work. Trying -3&4. FOIL shows this doesn’t work (but we are close, let’s try 3&-4). Trying 3&-4. This one works. Answer.

Note: The trial and error method may seem like an arduous task, but the more you practice the faster
Note also: The trial and error method is usually the better of the two methods to use if the leading
coefficient of the trinomial is a one.

IV. Expressions with four terms.

A. Group the expressions into two groups of two terms each.
B. Factor out the GCF of each group.
C. Recognize the common factor and use distributive property.

 Group the terms. We’ll try grouping the first two and the last two. Notice when we grouped the second two terms, we were careful to put the negative in front of the 𔄞uz” term inside the second parenthesis and put a plus sign in between the two sets of parentheses. If we group like this: (15z 2 + 5z) – (6uz – 2u) we have changed the original expression! Factor out GCFs for each group. Note that we could factor out a positive or negative 𔄚u” out of the second group. We factored out a negative, so that the signs for the binomial part in parentheses will match. Recognize (3z + 1) is a common factor and use the distributive property to write in factored form.

Note: If you try grouping the first two terms and last two terms and it doesn’t work, the commutative
property of addition allows us to try a different grouping (like the first and third in one group and
the second and fourth in the other). For an expression with four terms, there are three different
possible groupings.

Note also: You may be able to use the four-term grouping method for expressions with more than four
terms. For example, you may try grouping a five-term expression into a difference of squares
and a trinomial. Then apply the techniques discussed above for each of these groupings and
look for a common factor.

Note as well: Remember to factor completely. For instance, you may have to use difference of squares
more than once to get a completely factored form.

Note additionally: Factoring can be used to solve quadratic equations of the form ax 2 +bx+c = 0 (this is the
standard form of a quadratic equation). The process will be to set the quadratic equation
equal to zero (put it in standard form) and then factor it. Then you will use the zero-
product property, which states:
if AB=0, then A=0 or B=0 (or they both equal zero).

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## Example problems

Factor 1 2 p 7 − 1 8 p 2 − 3 0 12p^ <7>- 18p^ <2>- 30 1 2 p 7 − 1 8 p 2 − 3 0

Look for the greatest common factor of this polynomial using long division. You can do each of the terms separately or you can do them altogether like what we did here: GCF of multiple numbers

The GCF of this polynomial is found through multiplying the common factors from all three of the numbers together. This means you&aposll get:

We then factor out 6 from each term of the polynomial, and we&aposll get the final answer of:

Factor 1 0 z ( x + 2 y ) − 6 ( x + 2 y ) 10z(x + 2y) - 6(x + 2y) 1 0 z ( x + 2 y ) − 6 ( x + 2 y )

Firstly, let us look for the common factors of the polynomial. When you first look at the numbers, you&aposll likely spot out that the common factor of 10 and 6 is 2.

Another common factor is (x+2y). So, we factor out both and we&aposll get the final answer of:

If you ever need to double check your answer when trying to find the common factors of polynomials, try out this online GCF calculator. It will help you be sure of your answers when you factor more complex polynomials. As always, do remember that the calculator should only be used to check your answers rather than doing the questions for you!

Ready to move on? Learn how to complete the square in quadratic functions, convert quadratic functions from general to vertex form, and solve quadratic equations by factoring or by completing the square.

## Factoring - Math Notes

Common Factoring: Find out the GREATEST COMMON FACTOR of each term and factor it out. Using Grouping:
Sometimes, a polynomial will have no common factor for all the terms. Instead, we can group together the terms which have a common factor. When you use the Grouping Method:
* When there is no factor common to all terms
* When there is an even number of terms.
Example:

The polynomial x3+3x2−6x−18 has no single factor that is common to every term. However, we notice that if we group together the first two terms and the second two terms, we see that each resulting binomial has a particular factor common to both terms.

Factor x2 out of the first two terms, and factor −6 out of the second two terms. x2(x+3) −6(x+3)
Now look closely at this binomial. Each of the two terms contains the factor (x+3). Factor out (x+3).
(x+3) (x2−6) is the final factorization.
x3+3x2−6x−18= (x+3) (x2−6)

Notice that the first term in the resulting trinomial comes from the product of the first terms in the binomials: x⋅x=x2. The last term in the trinomial comes from the product of the last terms in the binomials: 4⋅7=28. The middle term comes from the addition of the outer and inner products: 7x+4x=11x. Also, notice that the coefficient of the middle term is exactly the sum of the last terms in the binomials: 4+7=11.

Method of Factoring
1. Write two sets of parentheses:( ) ( ).
2. Place a binomial into each set of parentheses. The first term of each binomial is a factor of the first term of the trinomial. 3. Determine the second terms of the binomials by determining the factors of the third term that when added together yield the coefficient of the middle term.

## Introduction to factoring polynomials Recall: Factors of a number are the numbers that divide the original number evenly.

Writing a number as a product of factors is called a factorization of the number.

The prime factorization of a number is the factorization of that number written as a product of prime numbers.

Common factors are factors that two or more numbers have in common.

The Greatest Common Factor (GCF) is the largest common factor.  The Greatest Common Factor of terms of a polynomial is the largest factor that the original terms share

The terms share a factor of x The terms share two factors of a

Note: The exponent of the variable in the GCF is the smallest exponent of that variable the terms To factor an expression means to write an equivalent expression that is a product

To factor a polynomial means to write the polynomial as a product of other polynomials

A factor that cannot be factored further is said to be a prime factor (prime polynomial)

A polynomial is factored completely if it is written as a product of prime polynomials

## Introduction to Factoring

Examples:
1. Find the GCF of 12 and 20.
2. Find the GCF of 48, 120, and 156.

3. Find the GCF of 12x^5 and 28x^3.

4. Find the GCF of  Homework: pg. 362 #5,9,13,17,19,21,25,27,31,37,39,41,43,45,51,53,59

4. 5x^2 + 40x + 60 (look for GCF)

8. –x^2 + 14x – 48 (look for GCF) Homework: pg. 369 #5-53 ever other odd (5,9,13,…)

Factoring Trinomials of the Form

Homework: pg. 377 #3-29 (odd), 31-65 (every other odd)

Factoring Special Binomials

Difference of Cubes : a^3 - b^3 = (a – b)(a^2 + ab + b^2)
Sum of Cubes : a^3 + b^3 = (a + b)(a^2 - ab + b^2)

Homework: pg. 384 #7-51 (every other odd), 53-87 (odd)

2. Determine the number of terms in the polynomial .
a. If there are only two terms, check to see if the binomial is one of the special binomials discussed in Section 6.4.
• Difference of Squares: a^2 – b^2 = (a+b)(a-b)
• Sum of Squares: a^2 + b^2 is not factorable
• Difference of Cubes: a^3 – b^3 = (a-b)(a^2+ab+b^2)
• Sum of Cubes: a^3 + b^3 = (a+b)(a^2-ab+b^2)

b. If there are three terms, try to factor the trinomial using the techniques of Sections 6.2 and 6.3.
• x^2 + bx + c = (x+m)(x+n): find two integers m and n whose product is c and whose sum is b.
• ax^2 + bx + c (a𕟁): factor using grouping, refer to object 1 in Section 6.3

c. If there are four terms, try factoring by grouping, discussed in Section 6.1.

3. After the polynomial has been factored, be sure that any factor with two or more terms does not have any common factors other than 1. If there are common factors, factor them out.

Homework: pg. 390 # 7-61 (every other odd)

If a*b=0, then either a = 0 or b = 0. 9. Find a quadratic equation that has the solutions -2 and -5.

Examples:
1. For f(x) = x^2 – 9x + 12, find f(-6).

2. Let f(x) = x^2 – 17x + 72. Find all values for x so that f(x) = 0.

3. Let f(x) = x^2 – 6x + 20. Find all values for x so that f(x) = 92.

Homework: pg. 404 #3,5,7,9,21,23,25,27

1. The product of two consecutive positive integers in 132. Find the two integers

2. The product of two consecutive even negative integers is 80. Find the two integers.

3. One positive number is 9 more than a second number, and their product is 112. Find the two numbers.

4. One positive number is 3 more than twice a second number, and their product is 189. Find the two numbers.

5. The area of a rectangle is 105 square feet . If the length of the rectangle is 8 feet more than its width, find the dimensions of the rectangle.

6. A homeowner poured a rectangular concrete slab in her backyard to use as a barbecue area. The length is 3 feet more than its width. There is a 2-foot wide flower bed around the barbecue area. If the area covered by the barbecue area and the flower bed is 270 square feet, find the dimensions of the barbecue area.
Dooley

## Mixed Practice

In the following exercises, factor.

 53. 54. 55. 56. 57. 58. ## MathHelp.com Previously, we have simplified expressions by distributing through parentheses, such as:

Simple factoring in the context of polynomial expressions is backwards from distributing. That is, instead of multiplying something through a parentheses and simplifying to get a polynomial expression, we will be seeing what we can take back out and put in front of a set of parentheses, such as undoing the multiplying-out that we just did above:

The trick in simple polynomial factoring is to figure what can be factored out of every term in the expression.

Warning: Don't make the mistake of thinking that "factoring" means "dividing something off and making it magically disappear". Remember that "factoring" means "dividing out of every term and moving it to be in front of the parentheses". Nothing "disappears" when we factor things merely get rearranged.

#### Factor 3x &ndash 12 .

The first term, the 3x , can be factored as (3)(x) the second term, the 12 , can be factored as (3)(4) . The only factor common to the two terms (that is, the only thing that can be divided out of each of the terms and then moved up in front of a set of parentheses) is the 3 .

I'll move this common factor out to the front. First, I'll write the common factor, and then draw an open-parenthesis:

When I divided the 3 out of the 3x , I was left with only the x remaining. I'll put that x as my first term inside the parentheses:

When I divided the 3 out of the &ndash12 , I left a &ndash4 behind, so I'll put that in the parentheses, too, followed by an end-parenthesis:

This factored form is my final answer:

Be careful not to drop "minus" signs when you factor.

Some books teach this topic by using the concept of the Greatest Common Factor, or GCF. In that case, you would methodically find the GCF of all the terms in the expression, put this in front of the parentheses, and then divide each term by the GCF and put the resulting expression inside the parentheses. The result will be the same as what I did above, and would look like this:

I divide the GCF out of each of the two terms:

Then I rewrite the expression in factored form, putting the GCF out in front, with the after-division values inside a parenthetical:

But the above process usually seems like an awful lot of work to me, so I usually just go straight to the factoring.

#### Factor 7x &ndash 7.

Looking at the expression they've given me, I see that I can usefully factor the two terms as (7)(x) and (7)(&ndash1) . In particular, this tells me that I can factor a 7 out of each of the terms. I'll factor this 7 out front, and start my parenthetical:

Dividing the 7 out of 7x leaves just an x , which I'll put inside the beginning of my parenthetical:

What am I left with when I divide the 7 out of the second term? I am not left with "nothing"! In fact, division of &ndash7 by 7 leaves me with &ndash1 (as I'd shown in my factorization above). This allows me to complete my parenthetical:

Take careful note: When you might think that "nothing" is left after factoring, it's usually the case that a " 1 " of some sort is left behind to go inside the parentheses.

#### Factor 12y 2 &ndash 5y .

In the expression they've given me, no number is a common factor of the two terms that is, the constants of the two terms, the 12 and the 5 , share no common numerical factors. But that doesn't mean that I can't factoring anything at all. I can still factor out a common variable.

In this case, I can pull a factor of y from each of the two terms, using the fact that 12y 2 can be restated as (12y)(y) , and &ndash5y can be restated as (&ndash5)(y) .

Putting the common (variable) factor out in front of an open-paren, I have:

In the first term of the original expression, after dividing out one copy of y , I have 12y left over. This goes in the beginning of my parenthetical:

(This is what is left to go inside the parenthetical because 12y 2 means 12×y×y , so taking the 12 and one of the y 's out front leaves the second y behind.)

Looking at the second term of the original expression, after I factor out the y , I have the &ndash5 left over. This finishes my parenthetical, and my answer is: