Algebra 1 Interactive Chalkboard
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GLENCOE DIVISION
Glencoe/McGraw-Hill
8787 Orion Place
Columbus, Ohio 43240
Lesson 9-1 Factors and Greatest Common Factors
Lesson 9-2 Factoring Using the Distributive Property
Lesson 9-3 Factoring Trinomials: x2 + bx + c
Lesson 9-4 Factoring Trinomials: ax2 + bx + c
Lesson 9-5 Factoring Differences of Squares
Lesson 9-6 Perfect Squares and Factoring
Example 1 Classify Numbers as Prime or Composite
Example 2 Prime Factorization of a Positive Integer
Example 3 Prime Factorization of a Negative Integer
Example 4 Prime Factorization of a Monomial
Example 5 GCF of a Set of Monomials
Example 6 Use Factors
Factor 22. Then classify it as prime or composite.
To find the factors of 22, list all pairs of whole numbers
whose product is 22.
Answer: Since 22 has more than two factors, it is
a composite number. The factors of 22, in
increasing order, are 1, 2, 11, and 22.
Factor 31. Then classify it as prime or composite.
The only whole numbers that can be multiplied together
to get 31 are 1 and 31.
Answer: The factors of 31 are 1 and 31. Since the
only factors of 31 are 1 and itself, 31 is a
prime number.
Factor each number. Then classify it as prime
or composite.
a. 17
Answer: 1, 17; prime
b. 25
Answer: 1, 5, 25; composite
Find the prime factorization of 84.
Method 1
The least prime factor of 84 is 2.
The least prime factor of 42 is 2.
The least prime factor of 21 is 3.
All of the factors in the last row are prime.
Answer: Thus, the prime factorization of 84 is
Method 2 Use a factor tree.
84
21
3
4
7
2
2
and
All of the factors in the last branch of the factor tree
are prime.
Answer: Thus, the prime factorization of 84 is
or
Find the prime factorization of 60.
Answer:
or
Find the prime factorization of –132.
Express –132 as –1 times 132.
/ \
/ \
/ \
Answer: The prime factorization of –132 is
or
Find the prime factorization of –154.
Answer:
Factor
Answer:
completely.
in factored form is
Factor
completely.
Express –26 as –1 times 26.
Answer:
in factored form is
Factor each monomial completely.
a.
Answer:
b.
Answer:
Find the GCF of 12 and 18.
Factor each number.
Circle the common prime factors.
The integers 12 and 18 have one 2 and one 3 as
common prime factors. The product of these common
prime factors,
or 6, is the GCF.
Answer: The GCF of 12 and 18 is 6.
.
Find the GCF of
Factor each number.
Circle the common prime factors.
Answer: The GCF of
and
is
.
Find the GCF of each set of monomials.
a. 15 and 35
Answer: 5
b.
Answer:
and
Crafts Rene has crocheted 32 squares for an afghan.
Each square is 1 foot square. She is not sure how she
will arrange the squares but does know it will be
rectangular and have a ribbon trim. What is the
maximum amount of ribbon she might need to finish
an afghan?
Find the factors of 32 and draw rectangles with each
length and width. Then find each perimeter.
The factors of 32 are 1, 2, 4, 8, 16, 32.
The greatest perimeter is 66 feet. The afghan with this
perimeter has a length of 32 feet and a width of 1 foot.
Answer: The maximum amount of ribbon Rene will need
is 66 feet.
Mary wants to plant a rectangular
flower bed in her front yard with a
stone border. The area of the flower
bed will be 45 square feet and the
stones are one foot square each. What
is the maximum number of stones that
Mary will need to go around all four
sides of the flower bed?
Answer: 92 feet
Example 1 Use the Distributive Property
Example 2 Use Grouping
Example 3 Use the Additive Inverse Property
Example 4 Solve an Equation in Factored Form
Example 5 Solve an Equation by Factoring
Use the Distributive Property to factor
First, find the CGF of 15x and
.
Factor each number.
.
Circle the common prime factors.
GFC:
Write each term as the product of the GCF and its
remaining factors. Then use the Distributive Property
to factor out the GCF.
Rewrite each term using
the GCF.
Simplify remaining factors.
Distributive Property
Answer: The completely factored form of
is
Use the Distributive Property to factor
.
Factor each number.
Circle the common prime factors.
GFC:
or
Rewrite each term using the GCF.
Distributive Property
Answer: The factored form of
is
Use the Distributive Property to factor each polynomial.
a.
Answer:
b.
Answer:
Factor
Group terms with
common factors.
Factor the GCF
from each grouping.
Answer:
Distributive Property
Factor
Answer:
Factor
Group terms with common factors.
Factor GCF from each grouping.
Answer:
Distributive Property
Factor
Answer:
Solve
Then check the solutions.
If
Property either
, then according to the Zero Product
or
Original equation
or
Set each factor equal to zero.
Solve each equation.
Answer: The solution set is
Check Substitute 2 and
for x in the original equation.
Solve
Answer: {3, –2}
Then check the solutions.
Solve
Then check the solutions.
Write the equation so that it is of the form
Original equation
Subtract
from each side.
Factor the GCF of 4y and
which is 4y.
or
Zero Product Property
Solve each equation.
Answer: The solution set is
0 and
Check by substituting
for y in the original equation.
Solve
Answer:
Example 1 b and c Are Positive
Example 2 b Is Negative and c Is Positive
Example 3 b Is Positive and c Is Negative
Example 4 b Is Negative and c Is Negative
Example 5 Solve an Equation by Factoring
Example 6 Solve a Real-World Problem by Factoring
Factor
In this trinomial,
and
You need to find the two
numbers whose sum is 7 and whose product is 12. Make
an organized list of the factors of 12, and look for the pair
of factors whose sum is 7.
Factors of 12 Sum of Factors
1, 12
2, 6
3, 4
Answer:
13
8
7
The correct factors are
3 and 4.
Write the pattern.
and
Check You can check the result by multiplying the
two factors.
F
O
I
L
FOIL method
Simplify.
Factor
Answer:
Factor
In this trinomial,
and
This means
is
negative and mn is positive. So m and n must both be
negative. Therefore, make a list of the negative factors of
27, and look for the pair whose sum is –12.
Factors of 27 Sum of Factors
–1, –27
–3, –9
Answer:
–28
–12
The correct factors are
–3 and –9.
Write the pattern.
and
Check You can check this result by using a graphing
calculator. Graph
and
on the same screen. Since only one graph appears,
the two graphs must coincide. Therefore, the trinomial
has been factored correctly.
Factor
Answer:
Factor
In this trinomial,
and
This means
is
positive and mn is negative, so either m or n is negative,
but not both. Therefore, make a list of the factors of –18
where one factor of each pair is negative. Look for the pair
of factors whose sum is 3.
Factors of –18
1, –18
–1, 18
2, –9
–2, 9
3, –6
–3, 6
Sum of Factors
–17
17
– 7
7
The correct factors
– 3
are –3 and 6.
3
Write the pattern.
Answer:
and
Factor
Answer:
Factor
Since
and
is negative and mn is
negative. So either m or n is negative, but not both.
Factors of –20 Sum of Factors
1, –20
–1, 20
2, –10
–2, 10
4, –5
–4, 5
–19
19
– 8
8
– 1
1
The correct factors are
4 and –5.
Answer:
Write the pattern.
and
Factor
Answer:
Solve
Check your solutions.
Original equation
Rewrite the equation so that
one side equals 0.
Factor.
or
Zero Product Property
Solve each equation.
Answer: The solution is
Check Substitute –5 and 3 for x in the original equation.
Solve
Answer:
Check your solutions.
Architecture Marion has a small art studio measuring
10 feet by 12 feet in her backyard. She wants to build a
new studio that has three times the area of the old
studio by increasing the length and width by the same
amount. What will be the dimensions of the new
studio?
Explore Begin by
making a diagram like the
one shown to the right,
labeling the appropriate
dimensions.
Plan
Let
the amount added to each dimension of
the studio.
The new length times the new width equals the new area.
old area
Solve
Write the equation.
Multiply.
Subtract 360 from
each side.
Factor.
or
Zero Product
Property
Solve each equation.
Examine The solution set is
Only 8 is a valid
solution, since dimensions cannot be negative.
Answer: The length of the new studio should be
or 20 feet and the new width should be
or 18 feet.
Photography Adina has a
photograph. She wants
to enlarge the photograph by increasing the length
and width by the same amount. What dimensions of
the enlarged photograph will be twice the area of the
original photograph?
Answer:
Example 1 Factor ax2 + bx + c
Example 2 Factor When a, b, and c Have a
Common Factor
Example 3 Determine Whether a Polynomial Is Prime
Example 4 Solve Equations by Factoring
Example 5 Solve Real-World Problems by Factoring
Factor
In this trinomial,
and
You need to
find two numbers whose sum is 27 and whose product is
or 50. Make an organized list of factors of 50 and
look for the pair of factors whose sum is 27.
Factors of 50 Sum of Factors
1, 50
2, 25
51
27
The correct factors are
2 and 25.
Write the pattern.
and
Group terms with
common factors.
Factor the GCF from
each grouping.
Distributive Property
Answer:
Check You can check this result by multiplying the two factors.
F
O
I
L
FOIL method
Simplify.
Factor
Answer:
Factor
In this trinomial,
and
Since b is
negative,
is negative. Since c is positive, mn is
positive. So m and n must both be negative. Therefore,
make a list of the negative factors of
or 72, and look
for the pair of factors whose sum is –22.
Factors of 72 Sum of Factors
–1, –72
–2, –36
–4, –24
–4, –18
–73
–38
–27
–22
The correct factors are
–4, –18.
Write the pattern.
and
Group terms with
common factors.
Factor the GCF from
each grouping.
Answer:
Distributive Property
a. Factor
Answer:
b. Factor
Answer:
Factor
Notice that the GCF of the terms
, and 32 is 4.
When the GCF of the terms of a trinomial is an integer
other than 1, you should first factor out this GCF.
Distributive Property
Now factor
Since the lead coefficient is 1, find
the two factors of 8 whose sum is 6.
Factors of 8 Sum of Factors
1, 8
2, 4
9
6
The correct factors are
2 and 4.
Answer: So,
complete factorization of
Thus, the
is
Factor
Answer:
Factor
In this trinomial,
and
Since b is
positive,
is positive. Since c is negative, mn is
negative, so either m or n is negative, but not both.
Therefore, make a list of all the factors of 3(–5) or –15,
where one factor in each pair is negative. Look for the pair
of factors whose sum is 7.
Factors of –15
Sum of Factors
–1, 15
1, –15
–3, 5
3, –5
14
–14
2
–2
There are no factors whose sum is 7. Therefore,
cannot be factored using integers.
Answer:
is a prime polynomial.
Factor
Answer: prime
Solve
Original equation
Rewrite so one side equals 0.
Factor the left side.
or
Zero Product Property
Solve each equation.
Answer: The solution set is
Solve
Answer:
Model Rockets Ms. Nguyen’s science class built an
air-launched model rocket for a competition. When
they test-launched their rocket outside the classroom,
the rocket landed in a nearby tree. If the launch pad
was 2 feet above the ground, the initial velocity of the
rocket was 64 feet per second, and the rocket landed
30 feet above the ground, how long was the rocket in
flight? Use the equation
Vertical motion model
Subtract 30 from each side.
Factor out –4.
Divide each side by –4.
Factor
or
Zero Product Property
Solve each equation.
The solutions are
and
seconds. The first time
represents how long it takes the rocket to reach a height of
30 feet on its way up. The second time represents how
long it will take for the rocket to reach the height of 30 feet
again on its way down. Thus the rocket will be in flight for
3.5 seconds before coming down again.
Answer: 3.5 seconds
When Mario jumps over a hurdle, his feet leave the
ground traveling at an initial upward velocity of 12 feet
per second. Find the time t in seconds it takes for
Mario’s feet to reach the ground again. Use the
equation
Answer:
second
Example 1 Factor the Difference of Squares
Example 2 Factor Out a Common Factor
Example 3 Apply a Factoring Technique More
Than Once
Example 4 Apply Several Different Factoring Techniques
Example 5 Solve Equations by Factoring
Example 6 Use Differences of Two Squares
Factor
.
Write in form
Answer:
Factor the difference
of squares.
Factor
.
and
Answer:
Factor the difference
of squares.
Factor each binomial.
a.
Answer:
b.
Answer:
Factor
The GCF of
and 27b is 3b.
and
Answer:
Factor the difference
of squares.
Factor
Answer:
Factor
The GCF of
and 2500 is 4.
and
Factor the difference
of squares.
and
Answer:
Factor the difference
of squares.
Factor
Answer:
Factor
Original Polynomial
Factor out the GCF.
Group terms with
common factors.
Factor each grouping.
is the
common factor.
Answer:
Factor the difference
of squares,
into
.
Factor
Answer:
Solve
by factoring. Check your solutions.
Original equation.
and
Factor the difference of squares.
or
Zero Product Property
Solve each equation.
Answer: The solution set is
Check each solution in the original equation.
Solve
by factoring. Check your solutions.
Original equation
Subtract 3y from each side.
The GCF of
and 3y is 3y.
and
Applying the Zero Product Property, set each factor equal
to zero and solve the resulting three equations.
or
or
Answer: The solution set is
Check each solution in the original equation.
Solve each equation by factoring. Check your solutions.
a.
Answer:
b.
Answer:
Extended-Response Test Item
A square with side length x is cut from a right triangle
shown below.
a. Write an equation in terms of x that
represents the area A of the figure
after the corner is removed.
b. What value of x will result in a figure
that is
the area of the original
triangle? Show how you arrived at
your answer.
Read the Test Item
A is the area of the triangle minus the area of the square
that is to be removed.
Solve the Test Item
a. The area of the triangle is
the area of the square is
or 64 square units and
square units.
Answer:
b. Find x so that A is
the area of the original triangle,
Translate the verbal statement.
and
Simplify.
Subtract 48 from each side.
Simplify.
Factor the difference of squares.
or
Zero Product Property
Solve each equation.
Answer: Since length cannot be negative, the only
reasonable solution is 4.
Extended-Response Test Item
A square with side length x is cut from the larger
square shown below.
a. Write an equation in terms of x that
represents the area A of the figure
after the corner is removed.
Answer:
b. What value of x will result in a figure
that is
of the area of the
original square?
Answer: 3
Example 1 Factor Perfect Square Trinomials
Example 2 Factor Completely
Example 3 Solve Equations with Repeated Factors
Example 4 Use the Square Root Property to
Solve Equations
Determine whether
is a perfect square
trinomial. If so, factor it.
Yes,
1. Is the first term a perfect square?
2. Is the last term a perfect square?
3. Is the middle term equal to
Answer:
Yes,
? Yes,
is a perfect square trinomial.
Write as
Factor using the pattern.
Determine whether
square trinomial. If so, factor it.
1. Is the first term a perfect square?
2. Is the last term a perfect square?
3. Is the middle term equal to
Answer:
is a perfect
Yes,
Yes,
? No,
is not a perfect square trinomial.
Determine whether each trinomial is a perfect square
trinomial. If so, factor it.
a.
Answer: not a perfect square trinomial
b.
Answer: yes;
Factor
.
First check for a GCF. Then, since the polynomial has two
terms, check for the difference of squares.
6 is the GCF.
and
Answer:
Factor the difference
of squares.
Factor
.
This polynomial has three terms that have a GCF of 1.
While the first term is a perfect square,
the last term is not. Therefore, this is not a perfect
square trinomial.
This trinomial is in the form
Are there two
numbers m and n whose product is
and whose sum is 8? Yes, the product of 20 and –12 is
–240 and their sum is 8.
Write the pattern.
and
Group terms with
common factors.
Factor out the GCF
from each grouping.
Answer:
is the
common factor.
Factor each polynomial.
a.
Answer:
b.
Answer:
Solve
Original equation
Recognize
as a perfect square trinomial.
Factor the perfect
square trinomial.
Set the repeated factor
equal to zero.
Solve for x.
Answer: Thus, the solution set is
Check this
solution in the original equation.
Solve
Answer:
Solve
.
Original equation
Square Root Property
Add 7 to each side.
or
Separate into two equations.
Simplify.
Answer: The solution set is
Check each
solution in the original equation.
Solve
.
Original equation
Recognize perfect
square trinomial.
Factor perfect
square trinomial.
Square Root Property
Subtract 6 from each side.
or
Separate into two equations.
Simplify.
Answer: The solution set is
Check this
solution in the original equation.
Solve
.
Original equation
Square Root Property
Subtract 9 from each side.
Answer: Since 8 is not a perfect square, the solution set is
Using a calculator, the approximate
solutions are
or about –6.17 and
or about –11.83.
Check You can check your answer using a graphing
calculator. Graph
and
Using the
INTERSECT feature of your graphing calculator, find
where
The check of –6.17 as one of the
approximate solutions is shown.
Solve each equation. Check your solutions.
a.
Answer:
b
Answer:
c.
Answer:
Explore online information about the
information introduced in this chapter.
Click on the Connect button to launch your browser
and go to the Algebra 1 Web site. At this site, you
will find extra examples for each lesson in the
Student Edition of your textbook. When you finish
exploring, exit the browser program to return to this
presentation. If you experience difficulty connecting
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