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Warm Up
Tell whether the second number is a factor
of the first number
1. 50, 6
no
2. 105, 7
3. List the factors of 28.
±14, ±28
yes
±1, ±2, ±4, ±7,
Tell whether each number is prime or
composite. If the number is composite, write
it as the product of two numbers.
4. 11 prime
5. 98 composite; 49 • 2
Objectives
Write the prime factorization of
numbers.
Find the GCF of monomials.
Vocabulary
prime factorization
greatest common factor
The whole numbers that are multiplied to find a
product are called factors of that product. A
number is divisible by its factors.
You can use the factors of a number to write the
number as a product. The number 12 can be
factored several ways.
Factorizations of 12
•
•
•
•
•
•
•
The order of factors does not change the product,
but there is only one example below that cannot
be factored further. The circled factorization is
the prime factorization because all the factors
are prime numbers. The prime factors can be
written in any order, and except for changes in
the order, there is only one way to write the
prime factorization of a number.
Factorizations of 12
•
•
•
•
•
•
•
Remember!
A prime number has exactly two factors, itself
and 1. The number 1 is not prime because it only
has one factor.
Example 1: Writing Prime Factorizations
Write the prime factorization of 98.
Method 1 Factor tree
Method 2 Ladder diagram
Choose any two factors
Choose a prime factor of 98
of 98 to begin. Keep finding
to begin. Keep dividing by
factors until each branch
prime factors until the
ends in a prime factor.
quotient is 1.
98
2 98
7 49
2  49
7 7

7
7
1
98 = 2  7  7
98 = 2  7  7
The prime factorization of 98 is 2  7  7 or 2  72.
Example 2
Write the prime factorization of each number.
a. 40
40
2  20
2  10
2  5
40 = 23  5
The prime factorization
of 40 is 2  2  2  5 or
23  5.
b. 33
11 33
3
33 = 3  11
The prime factorization
of 33 is 3  11.
Example 3
Write the prime factorization of each number.
c. 49
d. 19
49
7  7
49 = 7  7
The prime factorization
of 49 is 7  7 or 72.
1 19
19
19 = 1  19
The prime factorization
of 19 is 1  19.
Factors that are shared by two or more whole
numbers are called common factors. The greatest
of these common factors is called the greatest
common factor, or GCF.
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 32: 1, 2, 4, 8, 16, 32
Common factors: 1, 2, 4
The greatest of the common factors is 4.
Example 4: Finding the GCF of Numbers
Find the GCF of each pair of numbers.
100 and 60
Method 1 List the factors.
factors of 100: 1, 2, 4,
5, 10, 20, 25, 50, 100
List all the factors.
factors of 60: 1, 2, 3, 4, 5,
6, 10, 12, 15, 20, 30, 60
Circle the GCF.
The GCF of 100 and 60 is 20.
Example 5: Finding the GCF of Numbers
Find the GCF of each pair of numbers.
26 and 52
Method 2 Prime factorization.
26 =
2  13
52 = 2  2  13
2  13 = 26
Write the prime
factorization of each
number.
Align the common
factors.
The GCF of 26 and 52 is 26.
Example 6
Find the GCF of each pair of numbers.
12 and 16
Method 1 List the factors.
factors of 12: 1, 2, 3, 4, 6, 12
List all the factors.
factors of 16: 1, 2, 4, 8, 16
Circle the GCF.
The GCF of 12 and 16 is 4.
Example 7
Find the GCF of each pair of numbers.
15 and 25
Method 2 Prime factorization.
15 = 1  3  5
25 = 1  5  5
1
5=5
Write the prime
factorization of each
number.
Align the common
factors.
You can also find the GCF of monomials that
include variables. To find the GCF of monomials,
write the prime factorization of each coefficient
and write all powers of variables as products.
Then find the product of the common factors.
Example 3A: Finding the GCF of Monomials
Find the GCF of each pair of monomials.
15x3 and 9x2
15x3 = 3  5  x  x  x
9x2 = 3  3  x  x
3
Write the prime factorization of
each coefficient and write
powers as products.
Align the common factors.
x  x = 3x2 Find the product of the common
factors.
The GCF of 3x3 and 6x2 is 3x2.
Example 3B: Finding the GCF of Monomials
Find the GCF of each pair of monomials.
8x2 and 7y3
Write the prime
factorization of each
8x2 = 2  2  2 
xx
coefficient and write
7y3 =
7
y  y  y powers as products.
Align the common
factors.
The GCF 8x2 and 7y is 1.
There are no
common factors
other than 1.
Helpful Hint
If two terms contain the same variable raised to
different powers, the GCF will contain that
variable raised to the lower power.
Example 8
Find the GCF of each pair of monomials.
18g2 and 27g3
18g2 = 2  3  3 
27g3 =
gg
Write the prime factorization
of each coefficient and
write powers as products.
3  3  3  g  g  g Align the common factors.
33
gg
Find the product of the
common factors.
The GCF of 18g2 and 27g3 is 9g2.
Example 9
Find the GCF of each pair of monomials.
16a6 and 9b
16a6 = 2  2  2  2  a  a  a  a  a  a
9b =
The GCF of 16a6 and 7b is 1.
Write the prime
factorization of
each coefficient
and write powers
as products.
33b
Align the common
factors.
There are no common factors
other than 1.
Example 10
Find the GCF of each pair of monomials.
8x and 7v2
8x = 2  2  2  x
7v2 =
7vv
The GCF of 8x and 7v2 is 1.
Write the prime factorization
of each coefficient and
write powers as products.
Align the common factors.
There are no common
factors other than 1.
Example 11 A cafeteria has 18 chocolate-milk cartons and 24 regular-milk
cartons. The cook wants to arrange the cartons with the same number of
cartons in each row. Chocolate and regular milk will not be in the same row.
How many rows will there be if the cook puts the greatest possible number
of cartons in each row?
The 18 chocolate and 24 regular milk cartons
must be divided into groups of equal size. The
number of cartons in each row must be a
common factor of 18 and 24.
Example 11 Continued
Factors of 18: 1, 2, 3, 6, 9, 18
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Find the common
factors of 18
and 24.
The GCF of 18 and 24 is 6.
The greatest possible number of milk cartons in
each row is 6. Find the number of rows of each type
of milk when the cook puts the greatest number of
cartons in each row.
Example 11 Continued
18 chocolate milk cartons
= 3 rows
6 containers per row
24 regular milk cartons
6 containers per row
= 4 rows
When the greatest possible number of types of
milk is in each row, there are 7 rows in total.
Example 12
Adrianne is shopping for a CD storage unit.
She has 36 CDs by pop music artists and 48
CDs by country music artists. She wants to put
the same number of CDs on each shelf without
putting pop music and country music CDs on
the same shelf. If Adrianne puts the greatest
possible number of CDs on each shelf, how
many shelves does her storage unit need?
The 36 pop and 48 country CDs must be divided into
groups of equal size. The number of CDs in each row
must be a common factor of 36 and 48.
Example 12 Continued
Find the common
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 factors of 36
and 48.
Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
The GCF of 36 and 48 is 12.
The greatest possible number of CDs on each shelf
is 12. Find the number of shelves of each type of
CDs when Adrianne puts the greatest number of
CDs on each shelf.
36 pop CDs
12 CDs per shelf
= 3 shelves
48 country CDs
12 CDs per shelf
= 4 shelves
When the greatest possible number of CD types
are on each shelf, there are 7 shelves in total.
Recall that the Distributive Property states that
ab + ac =a(b + c). The Distributive Property
allows you to “factor” out the GCF of the terms in
a polynomial to write a factored form of the
polynomial.
A polynomial is in its factored form when it is
written as a product of monomials and polynomials
that cannot be factored further. The polynomial
2(3x – 4x) is not fully factored because the terms
in the parentheses have a common factor of x.
Example 13 : Factoring by Using the GCF
Factor the polynomial. Check your answer.
2x2 – 4
2x2 = 2 
xx
4=22
2
2x2 – (2  2)
2(x2 – 2)
Check 2(x2 – 2)
2x2 – 4
Find the GCF.
The GCF of 2x2 and 4 is 2.
Write terms as products using the
GCF as a factor.
Use the Distributive Property to factor
out the GCF.
Multiply to check your answer.
The product is the original
polynomial.
Example 14 : Factoring by Using the GCF
Factor the polynomial. Check your answer.
8x3 – 4x2 – 16x
8x3 = 2  2  2 
x  x  x Find the GCF.
4x2 = 2  2 
xx
16x = 2  2  2  2  x
The GCF of 8x3, 4x2, and 16x is
4x.
22
x = 4x Write terms as products using
the GCF as a factor.
2x2(4x) – x(4x) – 4(4x)
Use the Distributive Property to
4x(2x2 – x – 4)
factor out the GCF.
Check 4x(2x2 – x – 4)
Multiply to check your answer.
The product is the original
8x3 – 4x2 – 16x 
polynomials.
Example 15 : Factoring by Using the GCF
Factor the polynomial. Check your answer.
–14x – 12x2
– 1(14x + 12x2)
14x = 2 
7x
12x2 = 2  2  3 
xx
2
–1[7(2x) + 6x(2x)]
–1[2x(7 + 6x)]
–2x(7 + 6x)
Both coefficients are
negative. Factor out –1.
Find the GCF.
2
The
GCF
of
14x
and
12x
x = 2x
is 2x.
Write each term as a product
using the GCF.
Use the Distributive Property
to factor out the GCF.
Example 16: Factoring by Using the GCF
Factor the polynomial. Check your answer.
–14x – 12x2
Check –2x(7 + 6x)
–14x – 12x2 
Multiply to check your answer.
The product is the original
polynomial.
Caution!
When you factor out –1 as the first step, be sure
to include it in all the other steps as well.
Example 17: Factoring by Using the GCF
Factor the polynomial. Check your answer.
3x3 + 2x2 – 10
3x3 = 3
2x2 =
10 =
 x  x  x Find the GCF.
2
xx
25
3x3 + 2x2 – 10
There are no common
factors other than 1.
The polynomial cannot be factored further.
Example 18
Factor the polynomial. Check your answer.
5b + 9b3
5b = 5 
b
9b = 3  3  b  b  b
b
5(b) + 9b2(b)
b(5 + 9b2)
Check b(5 + 9b2)
5b + 9b3 
Find the GCF.
The GCF of 5b and 9b3 is b.
Write terms as products using
the GCF as a factor.
Use the Distributive Property to
factor out the GCF.
Multiply to check your answer.
The product is the original
polynomial.
Example 19
Factor the polynomial. Check your answer.
9d2 – 82
9d2 = 3  3  d  d
82 =
9d2 – 82
Find the GCF.
222222
There are no common
factors other than 1.
The polynomial cannot be factored further.
Example 20
Factor the polynomial. Check your answer.
–18y3 – 7y2
– 1(18y3 + 7y2)
Both coefficients are negative.
Factor out –1.
18y3 = 2  3  3  y  y  y
Find the GCF.
7y2 = 7 
yy
y  y = y2 The GCF of 18y3 and 7y2 is y2.
–1[18y(y2) + 7(y2)]
–1[y2(18y + 7)]
–y2(18y + 7)
Write each term as a product
using the GCF.
Use the Distributive Property
to factor out the GCF..
Example 21
Factor each polynomial. Check your answer.
8x4 + 4x3 – 2x2
8x4 = 2  2  2  x  x  x  x
4x3 = 2  2  x  x  x
Find the GCF.
2x2 = 2 
xx
2
x  x = 2x2 The GCF of 8x4, 4x3 and –2x2 is 2x2.
4x2(2x2) + 2x(2x2) –1(2x2) Write terms as products using the
2x2(4x2 + 2x – 1)
Check 2x2(4x2 + 2x – 1)
8x4 + 4x3 – 2x2
GCF as a factor.
Use the Distributive Property to factor
out the GCF.
Multiply to check your answer.
The product is the original polynomial.
To write expressions for the length and width of a
rectangle with area expressed by a polynomial,
you need to write the polynomial as a product.
You can write a polynomial as a product by
factoring it.
Example 22
The area of a court for the game squash is
9x2 + 6x m2. Factor this polynomial to find
possible expressions for the dimensions of
the squash court.
A = 9x2 + 6x
= 3x(3x) + 2(3x)
= 3x(3x + 2)
The GCF of 9x2 and 6x is 3x.
Write each term as a product
using the GCF as a factor.
Use the Distributive Property to
factor out the GCF.
Possible expressions for the dimensions of the
squash court are 3x m and (3x + 2) m.
Example 23
The area of the solar panel on another
calculator is (2x2 + 4x) cm2. Factor this
polynomial to find possible expressions for the
dimensions of the solar panel.
A = 2x2 + 4x
= x(2x) + 2(2x)
= 2x(x + 2)
The GCF of 2x2 and 4x is 2x.
Write each term as a product
using the GCF as a factor.
Use the Distributive Property to
factor out the GCF.
Possible expressions for the dimensions of the solar
panel are 2x cm, and (x + 2) cm.
Sometimes the GCF of terms is a binomial. This
GCF is called a common binomial factor. You
factor out a common binomial factor the same
way you factor out a monomial factor.
Example 24: Factoring Out a Common Binomial
Factor
Factor each expression.
A. 5(x + 2) + 3x(x + 2)
5(x + 2) + 3x(x + 2)
(x + 2)(5 + 3x)
The terms have a common
binomial factor of (x + 2).
Factor out (x + 2).
B. –2b(b2 + 1)+ (b2 + 1)
–2b(b2 + 1) + (b2 + 1) The terms have a common
binomial factor of (b2 + 1).
–2b(b2 + 1) + 1(b2 + 1) (b2 + 1) = 1(b2 + 1)
(b2 + 1)(–2b + 1)
Factor out (b2 + 1).
Example 25: Factoring Out a Common Binomial
Factor
Factor each expression.
C. 4z(z2 – 7) + 9(2z3 + 1)
There are no common
– 7) +
+ 1)
factors.
The expression cannot be factored.
4z(z2
9(2z3
Example 26
Factor each expression.
a. 4s(s + 6) – 5(s + 6)
4s(s + 6) – 5(s + 6)
(4s – 5)(s + 6)
The terms have a common
binomial factor of (s + 6).
Factor out (s + 6).
b. 7x(2x + 3) + (2x + 3)
7x(2x + 3) + (2x + 3)
The terms have a common
binomial factor of (2x + 3).
7x(2x + 3) + 1(2x + 3) (2x + 1) = 1(2x + 1)
(2x + 3)(7x + 1)
Factor out (2x + 3).
Example 27
Factor each expression.
c. 3x(y + 4) – 2y(x + 4)
3x(y + 4) – 2y(x + 4)
There are no common
factors.
The expression cannot be factored.
d. 5x(5x – 2) – 2(5x – 2)
5x(5x – 2) – 2(5x – 2)
(5x – 2)(5x – 2)
(5x – 2)2
The terms have a common
binomial factor of (5x – 2 ).
(5x – 2)(5x – 2) = (5x – 2)2
You may be able to factor a polynomial by
grouping. When a polynomial has four terms,
you can make two groups and factor out the
GCF from each group.
Example 28: Factoring by Grouping
Factor each polynomial by grouping.
Check your answer.
6h4 – 4h3 + 12h – 8
(6h4 – 4h3) + (12h – 8) Group terms that have a common
number or variable as a factor.
2h3(3h – 2) + 4(3h – 2) Factor out the GCF of each
group.
2h3(3h – 2) + 4(3h – 2) (3h – 2) is another common
factor.
(3h – 2)(2h3 + 4)
Factor out (3h – 2).
Example 28 Continued
Factor each polynomial by grouping.
Check your answer.
Check (3h – 2)(2h3 + 4)
Multiply to check your
solution.
3h(2h3) + 3h(4) – 2(2h3) – 2(4)
6h4 + 12h – 4h3 – 8
6h4 – 4h3 + 12h – 8
The product is the original
polynomial.
Example 29: Factoring by Grouping
Factor each polynomial by grouping.
Check your answer.
5y4 – 15y3 + y2 – 3y
(5y4 – 15y3) + (y2 – 3y)
Group terms.
5y3(y – 3) + y(y – 3)
Factor out the GCF of
each group.
5y3(y – 3) + y(y – 3)
(y – 3) is a common factor.
(y – 3)(5y3 + y)
Factor out (y – 3).
Example 29 Continued
Factor each polynomial by grouping.
Check your answer.
5y4 – 15y3 + y2 – 3y
Check (y – 3)(5y3 + y)
y(5y3) + y(y) – 3(5y3) – 3(y) Multiply to check your
solution.
5y4 + y2 – 15y3 – 3y
5y4 – 15y3 + y2 – 3y 
The product is the
original polynomial.
Example 30
Factor each polynomial by grouping.
Check your answer.
6b3 + 8b2 + 9b + 12
(6b3 + 8b2) + (9b + 12)
Group terms.
2b2(3b + 4) + 3(3b + 4)
2b2(3b + 4) + 3(3b + 4)
Factor out the GCF of
each group.
(3b + 4) is a common
factor.
(3b + 4)(2b2 + 3)
Factor out (3b + 4).
Example 30 Continued
Factor each polynomial by grouping.
Check your answer.
6b3 + 8b2 + 9b + 12
Check (3b +
4)(2b2
+ 3)
Multiply to check your
solution.
3b(2b2) + 3b(3)+ (4)(2b2) + (4)(3)
6b3 + 9b+ 8b2 + 12
6b3 + 8b2 + 9b + 12
The product is the
original polynomial.
Example 31
Factor each polynomial by grouping.
Check your answer.
4r3 + 24r + r2 + 6
(4r3 + 24r) + (r2 + 6)
Group terms.
4r(r2 + 6) + 1(r2 + 6)
Factor out the GCF of
each group.
(r2 + 6) is a common
factor.
4r(r2 + 6) + 1(r2 + 6)
(r2 + 6)(4r + 1)
Factor out (r2 + 6).
Example 32 Continued
Factor each polynomial by grouping.
Check your answer.
Check (4r + 1)(r2 + 6)
4r(r2) + 4r(6) +1(r2) + 1(6) Multiply to check your
solution.
4r3 + 24r +r2 + 6
4r3 + 24r + r2 + 6
The product is the
original polynomial.
Helpful Hint
If two quantities are opposites, their sum is 0.
(5 – x) + (x – 5)
5–x+x–5
–x+x+5–5
0+0
0
Recognizing opposite binomials can help you factor
polynomials. The binomials (5 – x) and (x – 5) are
opposites. Notice (5 – x) can be written as –1(x – 5).
–1(x – 5) = (–1)(x) + (–1)(–5)
Distributive Property.
= –x + 5
Simplify.
=5–x
Commutative Property
of Addition.
So, (5 – x) = –1(x – 5)
Example 33: Factoring with Opposites
Factor 2x3 – 12x2 + 18 – 3x
2x3 – 12x2 + 18 – 3x
(2x3 – 12x2) + (18 – 3x)
2x2(x – 6) + 3(6 – x)
2x2(x – 6) + 3(–1)(x – 6)
2x2(x – 6) – 3(x – 6)
(x – 6)(2x2 – 3)
Group terms.
Factor out the GCF of
each group.
Write (6 – x) as –1(x – 6).
Simplify. (x – 6) is a
common factor.
Factor out (x – 6).
Example 34
Factor each polynomial. Check your answer.
15x2 – 10x3 + 8x – 12
(15x2 – 10x3) + (8x – 12)
5x2(3 – 2x) + 4(2x – 3)
Group terms.
Factor out the GCF of
each group.
5x2(3 – 2x) + 4(–1)(3 – 2x) Write (2x – 3) as –1(3 – 2x).
5x2(3 – 2x) – 4(3 – 2x)
(3 – 2x)(5x2 – 4)
Simplify. (3 – 2x) is a
common factor.
Factor out (3 – 2x).
Example 35
Factor each polynomial. Check your answer.
8y – 8 – x + xy
(8y – 8) + (–x + xy)
Group terms.
8(y – 1)+ (x)(–1 + y)
Factor out the GCF of
each group.
8(y – 1)+ (x)(y – 1)
(y – 1) is a common
factor.
Factor out (y – 1) .
(y – 1)(8 + x)
Lesson Quiz: Part I
Factor each polynomial. Check your answer.
1. 16x + 20x3
4x(4 + 5x2)
2. 4m4 – 12m2 + 8m 4m(m3 – 3m + 2)
Factor each expression.
3. 7k(k – 3) + 4(k – 3)
4. 3y(2y + 3) – 5(2y + 3)
(k – 3)(7k + 4)
(2y + 3)(3y – 5)
Lesson Quiz: Part II
Factor each polynomial by grouping. Check your
answer.
5. 2x3 + x2 – 6x – 3
(2x + 1)(x2 – 3)
6. 7p4 – 2p3 + 63p – 18
(7p – 2)(p3 + 9)
7. A rocket is fired vertically into the air at 40 m/s.
The expression –5t2 + 40t + 20 gives the
rocket’s height after t seconds. Factor this
expression. –5(t2 – 8t – 4)
Warm Up
Simplify.
1. 2(w + 1) 2w + 2
2. 3x(x2 – 4) 3x3 – 12x
Find the GCF of each pair of monomials.
3. 4h2 and 6h 2h
4. 13p and 26p5 13p