Decimal, they can use
feature to find that
ons are -3, -1, and 1.
can also press
nd find where the
Y1 and Y2 are the same.
10,195.2 = (x + 7)x(x - 1)
Real-World
Substitute.
Graph y1 = 10,195.2 and y2 = (x + 7) x(x - 1).
Use the Intersect feature of the calculator. When
y = 10,195.2, x < 20. So x + 7 < 27 and x - 1 < 19.
Connection
A2Ch0604 Solving Polynomial Equations
LE
Alternative Method
International regulations
specify the allowable
dimensions of pet
transportation carriers.
Goal
To solve polynomial equations by factoring
Ca. State
2
1
4.0
Students factor polynomials representing the sum and difference of two cubes.
Solving Equations by Standard
Factoring
Recall that a quadratic difference of squares has a special factoring pattern. A cubic
sum of cubes and difference of cubes also have special factoring patterns.
cam09a2_te_1101.qxd
Sum and Difference of
Key Concepts
Two Cubes
cam09a2_te_1101.qxd
Properties
04/25/07
06:05 pm
and solve x3 - 19x =
0. –5, –1, 4
1
1
(
(
)
)
with
.
1, 4, 16, 64, c
1
Each term is 4 times the preceding term. The next term is 64 ? 4, or 256.
There will be 256 segments in the next figure in the pattern.
D
E
E
3
Real-World
EXAMPLE
10 ft
Connection
8.5 ft
B
A
4
5
A
A
B
B
D
C
E
E
You can use a variable, such as a, with positive integer subscripts to represent the
terms in a sequence.
EXAMPLE
1st
term 2nd term
3rd term a
c
n - 1 term nth term
Generating
Sequence
T side of the
T Koch snowflake,
T replace each
T
a. ToT create one
a1
a
a3
c
an - 1
an
Draw
the first2 four figures
of the pattern.
.
c
Advanced Learners
L4
1
E
E
Real-World
EXAMPLE
Connection
Multiple Choice Suppose you drop a handball from a height of 10 ft. After the ball
hits the floor, it rebounds to 85% of its previous height. About how high will the
ball rebound after its fourth bounce?
E
D
C
B
8.5 ft
6.1 ft
5.2 ft
Original height of ball: 10 ft S
1st term
T
a1
16a ! 250a
Advanced Learners
2nd term 3rd term c
T
T
a2
a3
c
5
n - 1 term
T
an - 1
nth term
T
an
Lesson 11-1
English Learners
L4
10 ft
n + 1 term
T
an + 1
Have students compare the recursive formula in
Example 3 with the composition of a function with
itself in Lesson 7-6.
8.5 ft
6.1 ft
5.2 ft
Original height of ball: 10 ft S
1st term
2nd term
3rd term
c
T
a1
T
a2
T
a3
c
n - 1 term
nth term
n + 1 term
c
T
an
T
an + 1
c
T
an - 1
Lesson 11-1
Have students compare the recursive formula in
Example 3 with the composition of a function with
itself in Lesson 7-6.
EXAMPLE
3
(
Connection
to Topology
611
611
611
Mathematical Patterns
EL
Write a3 on the board. Point out that the 3 is called a
subscript because the number is written below the
variable. Note that sub means underneath. Ask: What
other words begin with the prefix sub? samples:
subway, subterranean
2
2a
)
= 10 ± 10i 3
8
06:05 pm
Factoring
611
!2, 1 ± i 3
1 !1 ± i 3
,
3
6
Factor x ! 6x ! 27
let u = x 2 ,
u 2 ! 6u ! 27
rewrite with u
4
with
2
.
Guided Instruction
Help students understand that
each time a new person is added
to the group, the number of new
calls will equal the number of
people who were already in the
group.
1, 4, 16, 64, c
c. Predict the next term of the sequence. Explain your choice.
1
Each term is 4 times the preceding term. The next term is 64 ? 4, or 256.
There will be 256 segments in the next figure in the pattern.
1 Describe the pattern formed. Find the next three terms.
a. 27, 34, 41, 48, c
b. 243, 81, 27, 9, c
Divide by 3; 3, 1, 13 .
Add 7; 55, 62, 69.
(u ! 9 ) (u + 3)
Sometimes you can find the next term in a sequence by using a pattern from the
terms that come before it.
2
A
B
B
C
C
E
(x
E
D
D
E
E
Test-Taking Tip
10 ft
When you have to
repeat a step several
times, be careful to
use the correct number
of steps.
)(
8.5 ft
2
6.1 ft
5.2 ft
EXAMPLE
CD, Online, or Transparencies
Additional Examples
1 a. Start with a square with
sides 1 unit long. On the right
side, add on a square of the same
size. Continue adding one square
at a time in this way. Draw the
first four figures of the pattern.
)
! 9 x2 + 3
Original height of ball: 10 ft S
After 1st bounce: 85% of 10 = 0.85(10) = 8.5 S
After 2nd bounce: 0.85(8.5) = 7.225 S
After 3rd bounce: 0.85(7.225) < 6.141 S
After 4th bounce: 0.85(6.141) < 5.220 S
b. Write the number of 1-unit
segments in each figure above as
a sequence. 4, 7, 10, 13, . . .
c. Predict the next term of the
sequence. Explain your choice.
16; Each term is 3 more than the
preceding term.
T
a1
2nd term 3rd term c
T
a2
T
a3
c
n - 1 term
nth term
n + 1 term
c
T
an - 1
T
an
T
an + 1
c
Lesson 11-1 Mathematical Patterns
Advanced Learners
L4
Have students compare the recursive formula in
Example 3 with the composition of a function with
itself in Lesson 7-6.
a height of 100 cm. It bounces
back to 80% of its previous
height. How high will it go after
its fifth bounce? about 32.8 cm
611
English Learners EL
Write a3 on the board. Point out that the 3 is called a
subscript because the number is written below the
variable. Note that sub means underneath. Ask: What
other words begin with the prefix sub? samples:
subway, subterranean
611
.
c. Predict the next term of the sequence. Explain your choice.
let u = x
Guided Instruction
Activity
Teaching Tip
)
1, 4, 16, 64, c
Help students understand that
each time a new person is added
to the group, the number of new
calls will equal the number of
people who were already in the
group.
1
2
Each term is 4 times the preceding term. The next term is 64 ? 4, or 256.
There will be 256 segments in the next figure in the pattern.
(
1 Describe the pattern formed. Find the next three terms.
a. 27, 34, 41, 48, c
b. 243, 81, 27, 9, c
Divide by 3; 3, 1, 13 .
Add 7; 55, 62, 69.
3x u ! 6u + 5
2
5
A
A
B
B
E
D
E
D
E
D
C
C
D
D
E
E
EXAMPLE
Real-World
Connection
)
Test-Taking Tip
rewrite with u
CD, Online, or Transparencies
3x ( u ! 1) ( u ! 5 )
10 ft
When you have to
repeat a step several
times, be careful to
use the correct number
of steps.
(
8.5 ft
6.1 ft
5.2 ft
)(
Original height of ball: 10 ft S
3x x 2 ! 1 x 2 ! 5
After 1st bounce: 85% of 10 = 0.85(10) = 8.5 S
After 2nd bounce: 0.85(8.5) = 7.225 S
After 3rd bounce: 0.85(7.225) < 6.141 S
After 4th bounce: 0.85(6.141) < 5.220 S
Connection
to Topology
Additional Examples
Multiple Choice Suppose you drop a handball from a height of 10 ft. After the ball
hits the floor, it rebounds to 85% of its previous height. About how high will the
ball rebound after its fourth bounce?
E
D
C
B
EXAMPLE
You may wish to tell students that
the Koch snowflake is an example
of a fractal. Fractals have been
the subject of much mathematical
research since the mid-twentieth
century.
Sometimes you can find the next term in a sequence by using a pattern from the
terms that come before it.
2
4
C
B
A
3
C
B
A
2
C
B
A
1
(
)
factor
1 a. Start with a square with
sides 1 unit long. On the right
side, add on a square of the same
size. Continue adding one square
at a time in this way. Draw the
first four figures of the pattern.
rewrite x = u
2
b. Write the number of 1-unit
segments in each figure above as
a sequence. 4, 7, 10, 13, . . .
c. Predict the next term of the
sequence. Explain your choice.
16; Each term is 3 more than the
preceding term.
)
3x ( x + 1) ( x ! 1) x 2 ! 5 factor again
The ball will rebound about 5.2 ft after the fourth bounce.
The correct choice is D.
rewrite x = u
2
2 a. About how high will the ball rebound after the seventh bounce?
b. After what bounce will the rebound height be less than 2 ft?
a. 3.2 ft
b. 10th bounce
You can use a variable, such as a, with positive integer subscripts to represent the
terms in a sequence.
2 Suppose you drop a ball from
a height of 100 cm. It bounces
back to 80% of its previous
height. How high will it go after
its fifth bounce? about 32.8 cm
Try : Factor Completely
1st term
T
a1
2nd term
T
a2
3rd term c
T
a3
c
n - 1 term
T
an - 1
nth term
T
an
Advanced Learners
L4
n + 1 term
T
an + 1
c
c
A x 4 + 7x 2 + 6
Lesson 11-1
2 Suppose you drop a ball from
You can use a variable, such as a, with positive integer subscripts to represent the
terms in a sequence.
1st term
When you apply the
construction from Example 1
to an equilateral triangle,
you form the Koch
snowflake.
factor
( x + 3) ( x ! 3) ( x 2 + 3) factor again
The ball will rebound about 5.2 ft after the fourth bounce.
The correct choice is D.
2 a. About how high will the ball rebound after the seventh bounce?
b. After what bounce will the rebound height be less than 2 ft?
a. 3.2 ft
b. 10th bounce
with
b. Write the number of segments in each figure above as a sequence.
Connection
to Topology
You may wish to tell students that
the Koch snowflake is an example
of a fractal. Fractals have been
the subject of much mathematical
research since the mid-twentieth
century.
Connection
Multiple Choice Suppose you drop a handball from a height of 10 ft. After the ball
hits the floor, it rebounds to 85% of its previous height. About how high will the
ball rebound after its fourth bounce?
E
D
C
B
E
D
C
B
A
D
C
B
A
D
C
B
A
5
Real-World
EXAMPLE
2. Teach
Generating a Sequence
Factor completely 3x 5 ! 18x 3 + 15x
3x x 4 ! 6x 2 + 5
Factor GCF
(
2. Teach
b. Write the number of segments in each figure above as a sequence.
When you apply the
construction from Example 1
to an equilateral triangle,
you form the Koch
snowflake.
Page 611
EXAMPLE
a. To create one side of the Koch snowflake, replace each
Draw the first four figures of the pattern.
Generating a Sequence
EXAMPLE
Activity
Teaching Tip
A
06:05 pm
1
a. To create one side of the Koch snowflake, replace each
Draw the first four figures of the pattern.
4
04/25/07
Page 611
Factoring
1
= 5 ± 5i 3
4
5 5 ± 5i 3
4
cam09a2_te_1101.qxd
04/25/07
( !10 )2 ! 4 ( 4 ) ( 25 )
2(4)
= 10 ± 100 ! 400 = 10 ± !300
8
8
CD, Online, or Transparencies
27x 3 ! 1 = 0
B
! ( !10 ) ±
!b ± bof cubes.
! 4ac
Rewrite the expression as the difference
=
Additional Examples
Try : Solve
A x3 + 8 = 0
3
)
611
Solutions : ! ,
Not factorable,
2
English
Learners
use
Quadratic
Formula EL
Have students write the names along with the general
forms for these special cases of factoring: sum of
cubes, difference of cubes, difference of squares, and
perfect square trinomials. Ask students how the
factored forms can help them remember their names.
4x 2 ! 10x + 25
English Learners
L4
Have students create polynomial equations by
multiplying factors with both real and complex roots.
Have pairs solve each other’s equations.
2
c
EL
2 Suppose you drop a ball from
a height of 100 cm. It bounces
back to 80% of its previous
height. How high will it go after
its fifth bounce? about 32.8 cm
You can use a variable, such as a, with positive integer subscripts to represent the
terms in a sequence.
1
(
)
2
Write a3 on the board. Point out that the 3 is called a
subscript because the number is written below the
variable. Note that sub means underneath. Ask: What
other words begin with the prefix sub? samples:
b. Write the number of 1-unit
segments in each figure above as
a sequence. 4, 7, 10, 13, . . .
c. Predict the next term of the
sequence. Explain your choice.
16; Each term is 3 more than the
preceding term.
2 a. About how high will the ball rebound after the seventh bounce?
b. After what bounce will the rebound height be less than 2 ft?
a. 3.2 ft
b. 10th bounce
cam09a2_te_1101.qxd
2
2a ( 2a ! 5 ) 4a + 10a + 25
2
subway, subterranean
x=!5
One solution
Polynomials and Polynomial Functions
2
Advanced Learners
2 Suppose you drop a ball from
a height of 100 cm. It bounces
back to 80% of its previous
height. How high will it go after
its fifth bounce? about 32.8 cm
2
c
Mathematical Patterns
After 1st bounce: 85% of 10 = 0.85(10) = 8.5 S
The ball will rebound about 5.2 ft after the fourth bounce.
The correct choice is D.
L4
(
)
w ( w + 3) w ! 3w + 9
w + 27w
4
After 2nd bounce: 0.85(8.5) = 7.225 S
Advanced Learners
b. Write the number of 1-unit
segments in each figure above as
a sequence. 4, 7, 10, 13, . . .
c. Predict the next term of the
sequence. Explain your choice.
16; Each term is 3 more than the
preceding term.
The ball will rebound about 5.2 ft after the fourth bounce.
The correct choice is D.
1 a. Start with a square with
sides 1 unit long. On the right
side, add on a square of the same
size. Continue adding one square
at a time in this way. Draw the
first four figures of the pattern.
After 3rd bounce: 0.85(7.225) < 6.141 S
After 4th bounce: 0.85(6.141) < 5.220 S
Chapter 6
(
After 1st bounce: 85% of 10 = 0.85(10) = 8.5 S
2x +8x53 = 1.0 (2x – 1)(4x
Solve
for
2 ±
3 Factor
2xx± 1)
Test-Taking Tip
When you have to
repeat a step several
times, be careful to
use the correct number
of steps.
334
)
Additional Examples
1 a. Start with a square with
sides 1 unit long. On the right
side, add on a square of the same
size. Continue adding one square
at a time in this way. Draw the
first four figures of the pattern.
After 3rd bounce: 0.85(7.225) < 6.141 S
After 4th bounce: 0.85(6.141) < 5.220 S
You may wish to tell students that
the Koch snowflake is an example
of a fractal. Fractals have been
the subject of much mathematical
research since the mid-twentieth
century.
The answer is choice A.
E
D
C
B
(
CD, Online, or Transparencies
Connection
After 2nd bounce: 0.85(8.5) = 7.225 S
8x + 125 = ( 2x
) + ( 5 ) Factor
= (x - 2)(x 2 + 2 x + (2) 2 ) Factor.
2
= ( 2x+ +2 x5+) 4)4x 2 !Simplify.
10x + 25
= (x - 2)(x
2
E
D
C
B
A
10 ft
)
3
English Learners EL
Each term is 4 times the preceding term. The next term is 64 ? 4, or 256.
There will be 256 segments in the next figure in the pattern.
E
D
C
B
A
D
C
B
A
D
C
B
A
5
Test-Taking Tip
E
D
C
Real-World
EXAMPLE
You can use a variable, such as a, with positive integer subscripts to represent the
terms in a sequence.
Help students understand that
each time a new person is added
to the group, the number of new
calls will equal the number of
people who were already in the
group.
x 3 - 38 = (x) 3 - (2) 3
Have students compare the recursive formula in
Write a3 on the board. Point out that the 3 is called a
b. Write
the number
in each
figure
as ais sequence.
Example 3 with the composition
of a function
with of segments
subscript
because
theabove
number
written below the
variable. Note that sub means underneath. Ask: What
1, 4, 16, 64, c
other words begin with the prefix sub? samples:
subway,
subterranean
c. Predict the next term of the
sequence.
Explain your choice.
A
E
D
2 a. About how high will the ball rebound after the seventh bounce?
b. After what bounce will the rebound height be less than 2 ft?
a. 3.2 ft
b. 10th bounce
Activity
Teaching Tip
Sometimes you can find the next term in a sequence by using a pattern from the
terms that come before it.
4
E
D
Guided Instruction
611
1 Describe the pattern formed. Find the next three terms.
a. 27, 34, 41, 48, c
b. 243, 81, 27, 9, c
Divide by 3; 3, 1, 13 .
Add 7; 55, 62, 69.
3
Connection
to Topology
c
n + 1 term
T
with
an + 1
Lesson 11-1 Mathematical Patterns
When
youin
apply
the7-6.
itself
Lesson
construction from Example 1
to an equilateral triangle,
you form the Koch
snowflake.
2
E
D
When you have to
repeat a step several
times, be careful to
use the correct number
of steps.
2 Suppose you drop a ball from
a height of 100 cm. It bounces
back to 80% of its previous
height. How high will it go after
its fifth bounce? about 32.8 cm
b. After what bounce will the rebound height be less than 2 ft?
a. 3.2 ft
b. 10th bounce
1
1
EXAMPLE
Multiple
If you factor x3 - 8 in the form (x 2 A)(x2 1 Bx 12 C) , what is
factorable,
Solving Choice
a Polynomial
4x ! 10x + 25 Not
use Quadratic Formula
the value of A?
2. Teach
3
2
-2
4
-4
Solve 8x + 125 = 0
Test-Taking Tip
Remembering patterns
may help you find
answers more quickly.
E
D
C
)
(
06:05
pmAbout
Pagehow
611
2 a.
high will the ball rebound after the seventh bounce?
E
D
C
04/25/07
E
D
C
B
A
B
Multiple Choice Suppose you drop a handball from a height of 10 ft. After the ball
hits the floor, it rebounds to 85% of its previous height. About how high will the
ball rebound after its fourth bounce?
E
D
C
B
- a2b + ab2 + a2b - ab2 + b3
b. Write the number of 1-unit
segments in each figure above as
a sequence. 4, 7, 10, 13, . . .
c. Predict the next term of the
sequence. Explain your choice.
16; Each term is 3 more than the
preceding term.
The ball will rebound about 5.2 ft after the fourth bounce.
The correct choice is D.
3
A
B
1 a. Start with a square with
sides 1 unit long. On the right
side, add on a square of the same
size. Continue adding one square
at a time in this way. Draw the
first four figures of the pattern.
5.2 ft
After 1st bounce: 85% of 10 = 0.85(10) = 8.5 S
E
D
C
cam09a2_te_1101.qxd
E
D
C
B
A
D
C
B
A
1
You may wish to tell students that
the Koch snowflake is an example
of a fractal. Fractals have been
the subject of much mathematical
research since the mid-twentieth
century.
Sometimes you can find the next term in a sequence by using a pattern from the
terms that come before it.
Additional Examples
6.1 ft
Original height of ball: 10 ft S
After 2nd bounce: 0.85(8.5) = 7.225 S
2
Help students understand that
each time a new person is added
to the group, the number of new
calls will equal the number of
people who were already in the
group.
Each term is 4 times the preceding term. The next term is 64 ? 4, or 256.
There will be 256 segments in the next figure in the pattern.
CD, Online, or Transparencies
After 3rd bounce: 0.85(7.225) < 6.141 S
After 4th bounce: 0.85(6.141) < 5.220 S
1
5
Connection
to Topology
You may wish to tell students that
the Koch snowflake is an example
of a fractal. Fractals have been
the subject of much mathematical
research since the mid-twentieth
century.
Multiple Choice Suppose you drop a handball from a height of 10 ft. After the ball
hits the floor, it rebounds to 85% of its previous height. About how high will the
ball rebound after its fourth bounce?
Test-Taking Tip
When you have to
repeat a step several
times, be careful to
use the correct number
of steps.
Solving a Polynomial
)
Activity
Teaching Tip
1 Describe the pattern formed. Find the next three terms.
a. 27, 34, 41, 48, c
b. 243, 81, 27, 9, c
Divide by 3; 3, 1, 13 .
Add 7; 55, 62, 69.
Factor a + 64
B
=3 a3 + b3
3
a 3 + 64 = ( a ) + ( 4 )
C
= ( a +Factoring
4 ) a 2 ! 4a
+ 16or Difference of Cubes
a Sum
3 EXAMPLE
2
C
B
E
D
C
B
E
D
C
B
A
D
C
B
A
E
D
C
B
A
E
D
C
B
A
5
EXAMPLE
4
C
B
A
3
C
B
A
2
C
B
A
1
( =a
c. Predict the next term of the sequence. Explain your choice.
3
A
1, 4, 16, 64, c
2
Help students understand that
each time a new person is added
to the group, the number of new
calls will equal the number of
people who were already in the
group.
b. Write the number of segments in each figure above as a sequence.
Sometimes you can find the next term in a sequence by using a pattern from the
terms that come before it.
4
(
1 27 ! 8r 3
5
Guided Instruction
c. Predict the next term of the sequence. Explain your choice.
Guided Instruction
1 Describe the pattern formed. Find the next three terms.
a. 27, 34, 41, 48, c
b. 243, 81, 27, 9, c
Divide by 3; 3, 1, 13 .
Add 7; 55, 62, 69.
3
.
Factor x 3 ! 27
3
You
verify
! 27
= ( xthe
! ( 3) by multiplying. Here are steps for the sum
x 3 can
Tryof: cubes:
Factor Completely
)3 patterns
2) =
2 a(a2 - ab + b2) + b(a2 - ab + b2)
(a + b)(a2 =- ab
+
b
A 8x 3 ! 1
( 2x ! 1) 4x 2 + 2x + 1
( x ! 3) x + 3x + 9
When you apply the
construction from Example 1
to an equilateral triangle,
you form the Koch
snowflake.
2
27 ! 8r 3
5
5
3
3
1
= "#( 3) ! ( 2r ) $%
5
1
= ( 3 ! 2r ) 9 + 6r + 4r 2
5
with
b. Write the number of segments in each figure above as a sequence.
When you apply the
construction from Example 1
to an equilateral triangle,
you form the Koch
snowflake.
2. Teach
Generating a Sequence
EXAMPLE
2. Teach
Generating a Sequence
EXAMPLE
Factor Completely
a 3 + b 3 = ( a +a 3b+) ba32 =! (a
ab++b)(a
b 2 2 - ab + b 2 )
3
2
2
a 3 ! b 3 = ( a !a bb 2 + ab + b )
) ba32 =+ (aab-+b)(a
Page 611
Page 611
a. To create one side of the Koch snowflake, replace each
Draw the first four figures of the pattern.
Activity
Teaching Tip
1
06:05 pm
Sum and Difference of Cubes
a. To create one side of the Koch snowflake, replace each
Draw the first four figures of the pattern.
Ymin=–50
Ymax=50
Yscl=10
04/25/07
Sum and Difference of Cubes
ditional Examples
mensions in inches of the
ea inside a doghouse can
sed as width x, length
d height x - 3. The
15.9 ft3. Find the
ns of the inside of the
e. about 34 in. by 30 in.
p1
Intersection
X=19.954686 Y=10195.2
2 Find the dimensions of a carrier with volume 7 ft 3, width x inches,
length (x + 3) inches, and height (x - 2) inches.
22.7125 in., 25.7125 in., 20.7125 in.
ot need to multiply
x - 1). Have students
n its factored form and
the results are the
cuss the reasonable
with the students. Since
idth, the domain is
and it must be of a size
anageable to carry.
CD, Online, or Transparencies
The dimensions of the portable kennel are about 27 in.
by 20 in. by 19 in.
English Learners
Have students compare the recursive formula in
Example 3 with the composition of a function with
itself in Lesson 7-6.
B
Mathematical Patterns
611
EL
Write a3 on the board. Point out that the 3 is called a
subscript because the number is written below the
variable. Note that sub means underneath. Ask: What
other words begin with the prefix sub? samples:
4x ! 40x + 36x
5
subway, subterranean
3
(x
2
)(
)
+ 6 x2 + 1
4x ( x + 1) ( x ! 1) ( x + 3) ( x ! 3)
611
A2Ch0604 Solving Polynomial Equations
cam09a2_te_1101.qxd
04/25/07
06:05 pm
Solve by Factoring
1
2. Teach
Generating a Sequence
EXAMPLE
Solve x 4 = 4x 2 + 45
x 4 ! 4x 2 ! 45 = 0
u 2 ! 4u ! 45 = 0
(u + 5 )(u ! 9 ) = 0
a. To create one side of the Koch snowflake, replace each
Draw the first four figures of the pattern.
with
.
Guided Instruction
Activity
Teaching Tip
Help students understand that
each time a new person is added
to the group, the number of new
calls will equal the number of
people who were already in the
group.
b. Write the number of segments in each figure above as a sequence.
When you apply the
construction from Example 1
to an equilateral triangle,
you form the Koch
snowflake.
1, 4, 16, 64, c
c. Predict the next term of the sequence. Explain your choice.
1
Each term is 4 times the preceding term. The next term is 64 ? 4, or 256.
There will be 256 segments in the next figure in the pattern.
1 Describe the pattern formed. Find the next three terms.
a. 27, 34, 41, 48, c
b. 243, 81, 27, 9, c
Divide by 3; 3, 1, 13 .
Add 7; 55, 62, 69.
(
(x
4
5
A
A
B
B
E
D
E
D
C
C
D
D
E
E
Real-World
Connection
Multiple Choice Suppose you drop a handball from a height of 10 ft. After the ball
hits the floor, it rebounds to 85% of its previous height. About how high will the
ball rebound after its fourth bounce?
E
D
C
B
E
D
C
B
A
3
C
B
A
2
C
B
A
1
x2 + 5 x2 ! 9 = 0
EXAMPLE
Test-Taking Tip
10 ft
When you have to
repeat a step several
times, be careful to
use the correct number
of steps.
2
EXAMPLE
Connection
to Topology
You may wish to tell students that
the Koch snowflake is an example
of a fractal. Fractals have been
the subject of much mathematical
research since the mid-twentieth
century.
)(
)
+ 5 ) ( x + 3) ( x ! 3) = 0
Sometimes you can find the next term in a sequence by using a pattern from the
terms that come before it.
2
8.5 ft
6.1 ft
5.2 ft
Original height of ball: 10 ft S
CD, Online, or Transparencies
Additional Examples
1 a. Start with a square with
sides 1 unit long. On the right
side, add on a square of the same
size. Continue adding one square
at a time in this way. Draw the
first four figures of the pattern.
After 1st bounce: 85% of 10 = 0.85(10) = 8.5 S
After 2nd bounce: 0.85(8.5) = 7.225 S
Set to zero
rewrite with u = x
After 3rd bounce: 0.85(7.225) < 6.141 S
After 4th bounce: 0.85(6.141) < 5.220 S
b. Write the number of 1-unit
segments in each figure above as
a sequence. 4, 7, 10, 13, . . .
c. Predict the next term of the
sequence. Explain your choice.
16; Each term is 3 more than the
preceding term.
The ball will rebound about 5.2 ft after the fourth bounce.
The correct choice is D.
2 a. About how high will the ball rebound after the seventh bounce?
b. After what bounce will the rebound height be less than 2 ft?
a. 3.2 ft
b. 10th bounce
You can use a variable, such as a, with positive integer subscripts to represent the
terms in a sequence.
2nd term 3rd term c
T
T
a2
a3
c
2 Suppose you drop a ball from
a height of 100 cm. It bounces
back to 80% of its previous
height. How high will it go after
its fifth bounce? about 32.8 cm
c
x = ±3, ± i 5
1st term
T
a1
n - 1 term
T
an - 1
nth term
T
an
n + 1 term
T
an + 1
Lesson 11-1 Mathematical Patterns
L4
Have students compare the recursive formula in
Example 3 with the composition of a function with
itself in Lesson 7-6.
c
611
English Learners EL
Write a3 on the board. Point out that the 3 is called a
subscript because the number is written below the
variable. Note that sub means underneath. Ask: What
other words begin with the prefix sub? samples:
subway, subterranean
611
2
factor
rewrite with x
2
factor and solve
x2 ! 5 = 0 x + 3 = 0 x ! 3 = 0
x 2 = !5
x = !3
x=3
x = ±i 5
Advanced Learners
p2
Page 611
=u
Try : Solve
A x 4 + 11x 2 + 18 = 0
±3i, ± i 2
B
w 4 ! 7w 2 = !6
±1, ± 6