Transcript 7-1 - Garnet Valley

Factors
and
Greatest
and Greatest Common Factors
7-1
7-1 Factors
Common Factors
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Algebra 1Algebra 1
Holt
McDougal
7-1 Factors and Greatest Common Factors
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
Holt McDougal Algebra 1
5. 98 composite; 49  2
7-1 Factors and Greatest Common Factors
Objectives
Write the prime factorization of
numbers.
Find the GCF of monomials.
Holt McDougal Algebra 1
7-1 Factors and Greatest Common Factors
Vocabulary
prime factorization
greatest common factor
Holt McDougal Algebra 1
7-1 Factors and Greatest Common Factors
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

Holt McDougal Algebra 1






7-1 Factors and Greatest Common Factors
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

Holt McDougal Algebra 1






7-1 Factors and Greatest Common Factors
Remember!
A prime number has exactly two factors, itself
and 1. The number 1 is not prime because it only
has one factor.
Holt McDougal Algebra 1
7-1 Factors and Greatest Common Factors
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.
Holt McDougal Algebra 1
7-1 Factors and Greatest Common Factors
Check It Out! Example 1
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.
Holt McDougal Algebra 1
b. 33
11 33
3
33 = 3  11
The prime factorization
of 33 is 3  11.
7-1 Factors and Greatest Common Factors
Check It Out! Example 1
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.
Holt McDougal Algebra 1
1 19
19
19 = 1  19
The prime factorization
of 19 is 1  19.
7-1 Factors and Greatest Common Factors
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.
Holt McDougal Algebra 1
7-1 Factors and Greatest Common Factors
Example 2A: 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.
Holt McDougal Algebra 1
7-1 Factors and Greatest Common Factors
Example 2B: 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.
Holt McDougal Algebra 1
7-1 Factors and Greatest Common Factors
Check It Out! Example 2a
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.
Holt McDougal Algebra 1
7-1 Factors and Greatest Common Factors
Check It Out! Example 2b
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.
The GCF of 15 and 25 is 5.
Holt McDougal Algebra 1
7-1 Factors and Greatest 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.
Holt McDougal Algebra 1
7-1 Factors and Greatest 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 15x3 and 9x2 is 3x2.
Holt McDougal Algebra 1
7-1 Factors and Greatest Common Factors
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 7y3 is 1.
Holt McDougal Algebra 1
There are no
common factors
other than 1.
7-1 Factors and Greatest Common Factors
Helpful Hint
If two terms contain the same variable raised to
different powers, the GCF will contain that
variable raised to the lower power.
Holt McDougal Algebra 1
7-1 Factors and Greatest Common Factors
Check It Out! Example 3a
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.
Holt McDougal Algebra 1
7-1 Factors and Greatest Common Factors
Check It Out! Example 3b
Find the GCF of each pair of monomials.
Write the prime
factorization of
each coefficient
and write powers
as products.
16a6 and 9b
16a6 = 2  2  2  2  a  a  a  a  a  a
9b =
The GCF of 16a6 and 9b is 1.
Holt McDougal Algebra 1
33b
Align the common
factors.
There are no common factors
other than 1.
7-1 Factors and Greatest Common Factors
Check It Out! Example 3c
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.
Holt McDougal Algebra 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.
7-1 Factors and Greatest Common Factors
Example 4: Application
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.
Holt McDougal Algebra 1
7-1 Factors and Greatest Common Factors
Example 4 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.
Holt McDougal Algebra 1
7-1 Factors and Greatest Common Factors
Example 4 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.
Holt McDougal Algebra 1
7-1 Factors and Greatest Common Factors
Check It Out! Example 4
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.
Holt McDougal Algebra 1
7-1 Factors and Greatest Common Factors
Check It Out! Example 4 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.
Holt McDougal Algebra 1
7-1 Factors and Greatest Common Factors
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.
Holt McDougal Algebra 1
7-1 Factors and Greatest Common Factors
Lesson Quiz: Part I
Write the prime factorization of each number.
1. 50
2  52
2. 84
22  3  7
Find the GCF of each pair of numbers.
3. 18 and 75 3
4. 20 and 36 4
Holt McDougal Algebra 1
7-1 Factors and Greatest Common Factors
Lesson Quiz: Part II
Find the GCF of each pair of monomials.
5. 12x and 28x3 4x
6. 27x2 and 45x3y2 9x2
7. Cindi is planting a rectangular flower bed with 40
orange flower and 28 yellow flowers. She wants
to plant them so that each row will have the
same number of plants but of only one color. How
many rows will Cindi need if she puts the greatest
possible number of plants in each row?
17
Holt McDougal Algebra 1
and Greatest
7-1
7-2 Factors
Factoring
by GCF Common Factors
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Algebra 1Algebra 1
Holt
McDougal
7-1 Factors and Greatest Common Factors
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
Holt McDougal Algebra 1
7-1 Factors and Greatest Common Factors
Objective
Factor polynomials by using the
greatest common factor.
Holt McDougal Algebra 1
7-1 Factors and Greatest Common Factors
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.
Holt McDougal Algebra 1
7-1 Factors and Greatest Common Factors
Example 1A: Factoring by Using the GCF
Factor each polynomial. Check your answer.
2x2 – 4
2x2 = 2 
xx
4=22
Find the GCF.
2
2x2 – (2  2)
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.
2(x2 – 2)
Check 2(x2 – 2)
2x2 – 4
Holt McDougal Algebra 1
7-1 Factors and Greatest Common Factors
Writing Math
Aligning common factors can help you find the
greatest common factor of two or more terms.
Holt McDougal Algebra 1
7-1 Factors and Greatest Common Factors
Example 1B: Factoring by Using the GCF
Factor each 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.
Holt McDougal Algebra 1
7-1 Factors and Greatest Common Factors
Example 1C: Factoring by Using the GCF
Factor each 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)
Holt McDougal Algebra 1
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.
7-1 Factors and Greatest Common Factors
Example 1C: Continued
Factor each polynomial. Check your answer.
–14x – 12x2
Check –2x(7 + 6x)
–14x – 12x2 
Holt McDougal Algebra 1
Multiply to check your answer.
The product is the original
polynomial.
7-1 Factors and Greatest Common Factors
Caution!
When you factor out –1 as the first step, be sure
to include it in all the other steps as well.
Holt McDougal Algebra 1
7-1 Factors and Greatest Common Factors
Example 1D: Factoring by Using the GCF
Factor each 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.
Holt McDougal Algebra 1
7-1 Factors and Greatest Common Factors
Check It Out! Example 1a
Factor each 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 
Holt McDougal Algebra 1
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.
7-1 Factors and Greatest Common Factors
Check It Out! Example 1b
Factor each 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.
Holt McDougal Algebra 1
7-1 Factors and Greatest Common Factors
Check It Out! Example 1c
Factor each 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)
Holt McDougal Algebra 1
Write each term as a product
using the GCF.
Use the Distributive Property
to factor out the GCF..
7-1 Factors and Greatest Common Factors
Check It Out! Example 1d
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
Holt McDougal Algebra 1
GCF as a factor.
Use the Distributive Property to factor
out the GCF.
Multiply to check your answer.
The product is the original polynomial.
7-1 Factors and Greatest Common Factors
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.
Holt McDougal Algebra 1
7-1 Factors and Greatest Common Factors
Example 2: Application
The area of a court for the game squash is
(9x2 + 6x) square meters. 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.
Holt McDougal Algebra 1
7-1 Factors and Greatest Common Factors
Check It Out! Example 2
What if…? 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.
Holt McDougal Algebra 1
7-1 Factors and Greatest Common Factors
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.
Holt McDougal Algebra 1
7-1 Factors and Greatest Common Factors
Example 3: 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)
Holt McDougal Algebra 1
Factor out (b2 + 1).
7-1 Factors and Greatest Common Factors
Example 3: 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
Holt McDougal Algebra 1
9(2z3
7-1 Factors and Greatest Common Factors
Check It Out! Example 3
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 + 3) = 1(2x + 3)
(2x + 3)(7x + 1)
Holt McDougal Algebra 1
Factor out (2x + 3).
7-1 Factors and Greatest Common Factors
Check It Out! Example 3 : Continued
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
Holt McDougal Algebra 1
The terms have a common
binomial factor of (5x – 2 ).
(5x – 2)(5x – 2) = (5x – 2)2
7-1 Factors and Greatest Common Factors
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.
Holt McDougal Algebra 1
7-1 Factors and Greatest Common Factors
Example 4A: 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)
Holt McDougal Algebra 1
Factor out (3h – 2).
7-1 Factors and Greatest Common Factors
Example 4A 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
Holt McDougal Algebra 1
The product is the original
polynomial.
7-1 Factors and Greatest Common Factors
Example 4B: 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).
Holt McDougal Algebra 1
7-1 Factors and Greatest Common Factors
Example 4B 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 
Holt McDougal Algebra 1
The product is the
original polynomial.
7-1 Factors and Greatest Common Factors
Check It Out! Example 4a
Factor each polynomial by grouping.
Check your answer.
6b3 + 8b2 + 9b + 12
(6b3 + 8b2) + (9b + 12)
Group terms.
2b2(3b + 4) + 3(3b + 4)
Factor out the GCF of
each group.
(3b + 4) is a common
factor.
2b2(3b + 4) + 3(3b + 4)
(3b + 4)(2b2 + 3)
Holt McDougal Algebra 1
Factor out (3b + 4).
7-1 Factors and Greatest Common Factors
Check It Out! Example 4a 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
Holt McDougal Algebra 1

The product is the
original polynomial.
7-1 Factors and Greatest Common Factors
Check It Out! Example 4b
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)
Holt McDougal Algebra 1
Factor out (r2 + 6).
7-1 Factors and Greatest Common Factors
Check It Out! Example 4b 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
Holt McDougal Algebra 1
The product is the
original polynomial.
7-1 Factors and Greatest Common Factors
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
Holt McDougal Algebra 1
7-1 Factors and Greatest Common Factors
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)
Holt McDougal Algebra 1
7-1 Factors and Greatest Common Factors
Example 5: Factoring with Opposites
Factor 2x3 – 12x2 + 18 – 3x by grouping.
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)
Holt McDougal Algebra 1
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).
7-1 Factors and Greatest Common Factors
Check It Out! Example 5a
Factor each polynomial by grouping.
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)
Holt McDougal Algebra 1
Simplify. (3 – 2x) is a
common factor.
Factor out (3 – 2x).
7-1 Factors and Greatest Common Factors
Check It Out! Example 5b
Factor each polynomial by grouping.
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)
Holt McDougal Algebra 1
7-1 Factors and Greatest Common Factors
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)
Holt McDougal Algebra 1
(k – 3)(7k + 4)
(2y + 3)(3y – 5)
7-1 Factors and Greatest Common Factors
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)
Holt McDougal Algebra 1