9.1
Add and Subtract Polynomials
Goal
Your Notes
p Add and subtract polynomials.
VOCABULARY
Monomial
Degree of a monomial
Polynomial
Degree of a polynomial
Leading coefficient
Binomial
Trinomial
Example 1
Rewrite a polynomial
Write 7 1 2x 4 2 4x so that the exponents decrease
from left to right. Identify the degree and leading
coefficient of the polynomial.
Solution
Consider the degree of each of the polynomial's terms.
Degree is
.
Degree is
.
Degree is
.
7 1 2x 4 2 4x
The polynomial can be written as
. The
greatest degree is
, so the degree of the polynomial
is
, and the leading coefficient is
.
218
Lesson 9.1 • Algebra 1 Notetaking Guide
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Your Notes
Checkpoint Write the polynomial so that the
exponents decrease from left to right. Identify the
degree and leading coefficient of the polynomial.
1. 5x 1 13 1 8x 3
2. 4y 4 2 7y 5 1 2y
Example 2
Identify and classify polynomials
Tell whether the expression is a polynomial. If it is a
polynomial, find its degree and classify it by the number
of terms. Otherwise, tell why it is not a polynomial.
Expression
Is it a polynomial?
a. 26
Classify by degree
and number of terms
0 degree monomial
b. m23 1 4
c. 2h3 1 4h2
Yes
d. 9 2 5x 4 1 3x Yes
e. 2w 3 1 4w
Checkpoint Tell whether the expression is a polynomial.
If it is a polynomial, find its degree and classify it by
the number of terms. Otherwise, tell why it is not a
polynomial.
3. 4x 2 x7 1 5x 3
Copyright © Holt McDougal. All rights reserved.
4. v 3 1 v22 1 2v
Lesson 9.1 • Algebra 1 Notetaking Guide
219
Your Notes
Example 3
Add polynomials
Find the sum (a) (4x 3 1 x 2 2 5) 1 (7x 1 x 3 2 3x 2)
and (b) (x 2 1 x 1 8) 1 (x 2 2 x 2 1).
If a particular
power of the
variable appears in
one polynomial but
not the other, leave
a space in that
column, or write
the term with a
coefficient of 0.
Solution
a. Vertical format: Align like
4x3 1 x2
25
terms in vertical columns. 1 x3 2 3x2 1 7x
b. Horizontal format: Use the associative and
commutative properties to group like terms. then
simplify
(x 2 1 x 1 8) 1 (x 2 2 x 2 1)
)1(
5(
)1(
)
5
Example 4
Subtract polynomials
Find the difference (a) (4z 2 2 3) 2 (22z 2 1 5z 2 1)
and (b) (3x 2 1 6x 2 4) 2 (x 2 2 x 2 7).
Remember to
multiply each term
in the polynomial
by 21 when
you write the
subtraction as
addition.
Solution
( 4z 2
a.
2 3)
2 (22z 2 1 5z 2 1)
4z 2
2z 2
2 3
5z
1
b. (3x 2 1 6x 2 4) 2 (x 2 2 x 2 7)
5 3x 2 1 6x 2 4
5
5
Checkpoint Find the sum or difference.
Homework
5. (3x 4 2 2x 2 2 1) 1 (5x 3 2 x 2 1 9x 4)
6. (3t2 2 5t 1 t 4) 2 (11t 4 2 3t2)
220 Lesson 9.1 • Algebra 1 Notetaking Guide
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9.2
Multiply Polynomials
Goal
p Multiply polynomials.
Multiply a monomial and a polynomial
Example 1
Your Notes
Find the product 3x 3(2x 3 2 x 2 2 7x 2 3).
Solution
3x 3(2x 3 2 x 2 2 7x 2 3)
5 3x 3(
5
) 2 3x 3(
2
) 2 3x 3(
2
) 2 3x 3(
)
2
Multiply polynomials vertically and horizontally
Example 2
Find the product.
a. (a2 2 6a 2 3)(2a 2 5)
Remember that the
terms of (2a 2 5)
are 2a and 25.
They are not 2a
and 5.
Solution
a. Vertical format:
a2 2
3
2
a3 2
b. (3b2 2 2b 1 5)(5b 2 6)
6a 2 3
2a 2 5
Write the product
in vertical format.
a2 1
a1
Multiply by
a2 2
a
Multiply by
.
.
Add products.
b. Horizontal format:
(3b2 2 2b 1 5)(5b 2 6)
5
(5b 2 6) 2
(5b 2 6)
1
(5b 2 6)
5
5
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Lesson 9.2 • Algebra 1 Notetaking Guide
221
Your Notes
Checkpoint Find the product.
1. 2x 2(x 3 2 5x 2 1 3x 2 7)
2. (a2 1 5a 2 4)(2a 1 3)
Example 3
Multiply binomials using the FOIL pattern
Find the product (2c 1 7)(c 2 9).
Solution
(2c 1 7)(c 2 9)
5 2c(
) 1 2c(
) 1 7(
) 1 7(
)
5
5
Checkpoint Complete the following exercise.
3. Find the product (m 1 3)(5m 2 4).
222
Lesson 9.2 • Algebra 1 Notetaking Guide
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Your Notes
Example 4
Multiply polynomials to find an area
Area The dimensions of a rectangle are x 1 4 and
x 1 5. Write an expression that represents the area
of the rectangle.
Solution
Area 5 length p width
5(
)(
Formula for area
of a rectangle
)
Substitute for
length and width.
5
Multiply binomials.
5
Combine like
terms.
CHECK Use a graphing calculator
to check your answer. Graph
y1 5
and
y2 5
in the
same viewing window. The graphs
, so the product of
x 1 4 and x 1 5 is
.
Checkpoint Complete the following exercise.
4. The dimensions of a rectangle are x 1 3 and
x 1 11. Write an expression that represents the
area of the rectangle.
Copyright © Holt McDougal. All rights reserved.
Lesson 9.2 • Algebra 1 Notetaking Guide
223
Your Notes
Example 5
Solve a multi-step problem
Walkway You are making a
a walkway around part of
your swimming pool. The
dimensions of the swimming
pool and walkway are shown
in the diagram.
x ft
18 ft
x ft
25 ft
• Write a polynomial that represents the area of the
swimming pool.
• What is the area of the swimming pool if the walkway
is 2 feet wide?
Solution
Step 1 Write a polynomial using the formula for the area
of a rectangle. The length is
. The width
is
.
Area 5
p
5
5
5
Step 2 Substitute
Area 5
for x and evaluate.
5
The area of the swimming pool is
.
Checkpoint Complete the following exercise.
Homework
224
5. Swimming Pool Your neighbor
x ft
has a walkway around his
entire pool as shown in
the diagram. The width of
22 ft
the walkway is the same
on every side. Write a
30 ft
polynomial that represents
the area of the pool. What is the area
of the pool if the walkway is 3 feet wide?
Lesson 9.2 • Algebra 1 Notetaking Guide
x ft
Copyright © Holt McDougal. All rights reserved.
9.3
Find Special Products of
Polynomials
Goal
Your Notes
p Use special product patterns to multiply
polynomials.
SQUARE OF A BINOMIAL PATTERN
Algebra
(a 1 b)2 5 a2
1 b2
(a 2 b)2 5 a2
1 b2
Example
(x 1 4)2 5 x2
1 16
(3x 2 2)2 5 9x2
Example 1
14
Use the square of a binomial pattern
Find the product.
Solution
When you use
special product
patterns, remember
that a and b can be
numbers, variables,
or variable
expressions.
a. (4x 1 3)2 5 (4x)2
1 32
5 16x 2
19
b. (3x 2 5y)2 5 (3x)2
1 (5y)2
5 9x 2
1 25y2
Checkpoint Find the product.
1. (x 1 9)2
2. (2x 2 7)2
3. (5r 1 s)2
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Lesson 9.3 • Algebra 1 Notetaking Guide
225
Your Notes
SUM AND DIFFERENCE PATTERN
Algebra
(a 1 b)(a 2 b) 5
2
2
Example
(x 1 4)(x 2 4) 5
2
2
Example 2
2
Use the sum and difference pattern
Find the product.
Solution
a. (n 1 3)(n 2 3) 5
2
2
5
2
2
2
Sum and
difference pattern
Simplify.
b. (4x 1 y)(4x 2 y) 5
2
2
2
Sum and
difference pattern
5
2
2
2
Simplify.
Example 3
Use special products and mental math
Use special products to find the product 17 p 23.
Solution
Notice that 17 is 3 less than
than
.
17 p 23 5 (
226
Lesson 9.3 • Algebra 1 Notetaking Guide
2 3)(
1 3)
while 23 is 3 more
Write as product.
5
Sum and difference
pattern
5
Evaluate powers.
5
Simplify.
Copyright © Holt McDougal. All rights reserved.
Your Notes
Checkpoint Complete the following exercises.
4. Find the product (z 1 6)(z 2 6).
5. Find the product (4x 1 3)(4x 2 3).
6. Find the product (x 1 5y)(x 2 5y).
7. Describe how you can use special products to find
392.
Copyright © Holt McDougal. All rights reserved.
Lesson 9.3 • Algebra 1 Notetaking Guide
227
Your Notes
Example 4
Solve a multi-step problem
Eye Color An offspring's eye color is determined by a
combination of two genes, one inherited from each
parent. Each parent has two color genes, and the
offspring has an equal chance of inheriting either one.
Parent
The gene B is for brown eyes,
Bb
and the gene b is for blue eyes.
B
b
Any gene combination with a B
B BB Bb
results in brown eyes. Suppose
Parent
each parent has the same gene
Bb
b Bb
bb
combination Bb. The Punnett
square shows the possible gene
combinations of the offspring and the resulting eye color.
• What percent of the possible gene combinations of the
offspring result in blue eyes?
• Show how you could use a polynomial to model the
possible gene combinations of the offspring.
Solution
Step 1 Notice that the Punnett square shows that
out
of 4, or
of the possible gene combinations
result in blue eyes.
Step 2 Model the gene from each parent with
. The possible gene of the offspring
can be modeled by
. Notice that
this product also represents the area of the
Punnett square.
5
5
The coefficients show that
of the possible
gene combinations will result in blue eyes.
Homework
Checkpoint Complete the following exercise.
8. Eye Color Look back at Example 4. What percent
of the possible gene combinations of the offspring
result in brown eyes?
228
Lesson 9.3 • Algebra 1 Notetaking Guide
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9.4
Solve Polynomial Equations
in Factored Form
Goal
Your Notes
p Solve polynomial equations.
VOCABULARY
Roots
Vertical motion model
ZERO-PRODUCT PROPERTY
Let a and b be real numbers. If ab 5 0, then
or
5 0.
50
Use the zero-product property
Example 1
Solve (x 2 5)(x 1 4) 5 0.
Solution
(x 2 5)(x 1 4) 5 0
50
Write original
equation.
50
or
property
x5
x5
or
The solutions of the equation are
Solve for x.
.
CHECK Substitute each solution into the original
equation to check.
(
2 5)(
1 4) 0 0
(
2 5)(
1 4) 0 0
00
00
50
Copyright © Holt McDougal. All rights reserved.
50
Lesson 9.4 • Algebra 1 Notetaking Guide
229
Your Notes
Example 2
Find the greatest common monomial factor
Factor out the greatest common monomial factor.
a. 16x 1 40y
b. 6x2 1 30x3
Solution
a. The GCF of 16 and 40 is
. The variables x and y
have
. So, the greatest common
monomial factor of the terms is
.
16x 1 40y 5
b. The GCF of 6 and 30 is
. The GCF of x2 and x3 is
. So, the greatest common monomial factor of the
terms is
.
6x2 1 30x3 5
Example 3
Solve an equation by factoring
Solve the equation.
a.
3x2 1 15x 5 0
Original equation
50
50
x5
Factor left side.
50
or
x5
or
Zero-product property
Solve for x.
The solutions of the equation are
To use the zeroproduct property,
you must write the
equation so that
one side is 0. For
this reason,
must be subtracted
from each side of
the equation.
9b2 5 24b
b.
50
b5
or
or
Original equation
50
Subtract
each side.
50
Factor left side.
50
b5
The solutions of the equation are
230
Lesson 9.4 • Algebra 1 Notetaking Guide
.
from
Zero-product property
Solve for b.
.
Copyright © Holt McDougal. All rights reserved.
Your Notes
Checkpoint Solve the equation.
1. (x 1 6)(x 2 3) 5 0
2. (x 2 8)(x 2 5) 5 0
Checkpoint Factor out the greatest common
monomial factor.
3. 10x2 2 24y2
4. 3t 6 1 8t 4
The vertical motion
model takes
into account the
effect of gravity
but ignores other,
less significant,
factors such as air
resistance.
VERTICAL MOTION MODEL
The height h (in feet) of a projectile can be modeled by
h 5 216t 2 1 vt 1 s
where t is the
(in seconds) the object has been
in the air, v is the
(in feet per
second), and s is the
(in feet).
Copyright © Holt McDougal. All rights reserved.
Lesson 9.4 • Algebra 1 Notetaking Guide
231
Your Notes
Solve a multi-step problem
Example 4
Fountain A fountain sprays water into the air with an initial
vertical velocity of 20 feet per second. After how many
seconds does it land on the ground?
Solution
Step 1 Write a model for the water's height above ground.
h 5 216t 2 1 vt 1 s
Vertical motion model
h 5 216t 2 1
v5
t1
h 5 216t 2 1
and s 5
Simplify.
Step 2 Substitute
for h. When the water lands, its
height above the ground is
feet. Solve for t.
The solution
t 5 0 means that
before the water is
sprayed, its height
above the ground is
0 feet.
5 216t 2 1
Substitute
5
Factor right side.
for h.
or
Zero-product property
or
Solve for t.
The water lands on the ground
is sprayed.
seconds after it
Checkpoint Complete the following exercises.
5. Solve d2 2 7d 5 0.
Homework
232
6. Solve 8b2 5 2b.
7. What If? In Example 4, suppose the initial vertical
velocity is 18 feet per second. After how many
seconds does the water land on the ground?
Lesson 9.4 • Algebra 1 Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
9.5
Factor x 2 1 bx 1 c
p Factor trinomials of the form x 2 1 bx 1 c.
Goal
Your Notes
FACTORING x2 1 bx 1 c
Algebra
x2 1 bx 1 c 5 (x 1 p)(x 1 q) provided
and
5 c.
Example
x2 1 6x 1 5 5 (
and
5 5.
Example 1
)(
5b
56
) because
Factor when b and c are positive
Factor x 2 1 10x 1 16.
Solution
Find two
factors of
Make an organized list.
Factors of
16 1
8,
81
41
5
5
5
The factors 8 and
have a sum of
the correct values of p and q.
x 2 1 10x 1 16 5 (x 1 8)(
CHECK
(x 1 8)(
Copyright © Holt McDougal. All rights reserved.
.
Sum of factors
16,
4,
whose sum is
, so they are
)
)5
Multiply.
5
Simplify.
Lesson 9.5 • Algebra 1 Notetaking Guide
233
Your Notes
Example 2
Factor when b is negative and c is positive
Factor a2 2 5a 1 6.
Solution
Because b is negative and c is positive, p and q
must
.
Factors of
Sum of factors
1(
)5
1(
)5
a2 2 5a 1 6 5 (
Example 3
)(
)
Factor when b is positive and c is negative
Factor y2 1 3y 2 10.
Solution
Because c is negative, p and q must
.
Factors of
Sum of factors
210,
210 1
10,
10 1
25,
25 1
5,
51
y 2 1 3y 2 10 5 (
5
5
5
5
)(
)
Checkpoint Factor the trinomial.
1. x 1 7x 1 12
234
Lesson 9.5 • Algebra 1 Notetaking Guide
2. x 1 9x 1 8
Copyright © Holt McDougal. All rights reserved.
Your Notes
Checkpoint Factor the trinomial.
3. x 1 12x 1 27
4. x 2 2 9x 1 20
5. y 2 1 4y 2 21
6. z 2 1 2z 2 24
Example 4
Solve a polynomial equation
Solve the equation x 2 1 7x 5 18.
x 2 1 7x 5 18
x 2 1 7x 2
Write original equation.
50
Subtract
side.
50
Factor left side.
or
Zero-product property
or
Solve for x.
The solutions of the equation are
Copyright © Holt McDougal. All rights reserved.
from each
.
Lesson 9.5 • Algebra 1 Notetaking Guide
235
Your Notes
Solve a multi-step problem
Example 5
Dimensions The bandage shown
has an area of 16 square
centimeters. Find the width of
the bandage.
w cm
3 cm
w cm
3 cm
Solution
Step 1 Write an equation using the fact that the area of
the bandage is 16 square centimeters.
A5lpw
Formula for area
5
pw
05
Substitute values.
Simplify.
Step 2 Solve the equation for w.
05
Write equation.
05
Factor right side.
or
Zero-product
property
or
Solve for w.
The bandage cannot have a negative width, so the width
is
.
Checkpoint Complete the following exercises.
7. Solve the equation s2 2 12s 5 13.
Homework
236
8. What If? In Example 5, suppose the area of the
bandage is 27 square centimeters. What is the
width?
Lesson 9.5 • Algebra 1 Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
9.6
Factor ax 2 1 bx 1 c
Goal
p Factor trinomials of the form ax2 1 bx 1 c.
Factor when b is negative and c is positive
Example 1
Your Notes
Factor 2x 2 2 11x 1 5.
Solution
Because b is negative and c is positive, both factors of
c must be
. You must consider the
of the factors of 5, because the x-terms of the possible
factorizations are different.
Factors
of 2
Factors
of 5
Possible
factorization
Middle term when
multiplied
1, 2
21,
(x 2 1)(2x
)
2 2x 5
1, 2
25,
(x 2 5)(2x
)
2 10x 5
2x 2 2 11x 1 5 5 (x 2
Example 2
)(2x
)
Factor when b is positive and c is negative
Factor 5n2 1 2n 2 3.
Solution
Because b is positive and c is negative, the factors of
c have
.
Factors
of 5
Factors
of 23
Possible
factorization
Middle term when
multiplied
1, 5
1,
(n 1 1)(5n
)
1, 5
21,
(n 2 1)(5n
)
1, 5
3,
(n 1 3)(5n
)
1, 5
23,
(n 2 3)(5n
)
5n2 1 2n 2 3 5 (n
Copyright © Holt McDougal. All rights reserved.
)(5n
)
Lesson 9.6 • Algebra 1 Notetaking Guide
237
Your Notes
Checkpoint Factor the trinomial.
1. 3x 2 2 5x 1 2
Example 3
2. 2m2 1 m 2 21
Factor when a is negative
Factor 24x 2 1 4x 1 3.
Solution
Step 1 Factor
from each term of the trinomial.
24x 2 1 4x 1 3 5
(
Step 2 Factor the trinomial
and c are both
have
Factors
of 4
Remember to
include the
that you factored
out in Step 1.
Factors
of 23
)
. Because b
, the factors of c must
.
Possible
factorization
Middle term when
multiplied
1, 4
1,
(x 1 1)(4x
)
1, 4
3,
(x 1 3)(4x
)
1, 4
21,
(x 2 1)(4x
)
1, 4
23,
(x 2 3)(4x
)
2, 2
1,
(2x 1 1)(2x
)
2, 2
21,
(2x 2 1)(2x
)
24x 2 1 4x 1 3 5
Checkpoint Complete the following exercise.
3. Factor 22y 2 2 11y 2 5.
238
Lesson 9.6 • Algebra 1 Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Your Notes
Example 4
Write and solve a polynomial equation
Tennis An athlete hits a tennis ball at an initial height of
8 feet and with an initial vertical velocity of 62 feet per
second.
a. Write an equation that gives the height (in feet) of the
ball as a function of the time (in seconds) since it left
the racket.
b. After how many seconds does the ball hit the ground?
Solution
a. Use the
to write an equation
for the height h (in feet) of the ball.
h 5 216t 2 1 vt 1 s
h 5 216t 2 1
t1
v5
and s 5
b. To find the number of seconds that pass before the
ball lands, find the value of t for which the height
of the ball is
. Substitute
for h and solve the
equation for t.
5 216t 2 1
5
(
5
(
t1
Substitute
)
)(
for h.
Factor out
)
.
Factor the trinomial.
or
Zero-product property
or
Solve for t.
A negative solution does not make sense in this situation.
The tennis ball hits the ground after
.
Checkpoint Complete the following exercise.
Homework
4. What If? In Example 4, suppose another athlete
hits the tennis ball with an initial vertical velocity
of 20 feet per second from a height of 6 feet. After
how many seconds does the ball hit the ground?
Copyright © Holt McDougal. All rights reserved.
Lesson 9.6 • Algebra 1 Notetaking Guide
239
9.7
Factor Special Products
Goal
Your Notes
p Factor special products.
VOCABULARY
Perfect square trinomial
DIFFERENCE OF TWO SQUARES PATTERN
Algebra
a2 2 b2 5 (a 1 b)(
)
Example
9x2 2 4 5 (3x)2 2 22 5 (
Example 1
)(
)
Factor the differences of two squares
Factor the polynomial.
a. z2 2 81 5 z2 2
5 (z 1
2
)(z 2
b. 16x2 2 9 5 (
)
)2 2
5(
2
1
)(
c. a2 2 25b2 5 a2 2 (
)2
5 (a 1
d. 4 2 16n2 5
2
)(a 2
2
5
(
[(
)2 2 (
5
(
1
)
)
)
)2]
)(
2
)
Checkpoint Factor the polynomial.
1. x 2 2 100
2. 49y 2 2 25
3. c2 2 9d2
4. 45 2 80m2
240 Lesson 9.7 • Algebra 1 Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Your Notes
PERFECT SQUARE TRINOMIAL PATTERN
Algebra
a2 1 2ab 1 b2 5 (
)2
a2 2 2ab 1 b2 5 (
)2
Example
x2 1 8x 1 16 5 x2 1 2(x p 4) 1 42 5 (
)2
x2 2 6x 1 9 5 x2 2 2(x p 3) 1 32 5 (
Example 2
)2
Factor perfect square trinomials
Factor the polynomial.
a. x 2 2 16x 1 64 5 x 2 2 2(
5(
)2
b. 4y2 2 12y 1 9 5 (
)2 2 2(
5(
c. 9s2 1 6st 1 t2 5 (
2
)1
2
)2
)2 1 2(
5(
d. 23z 2 1 24z 2 48 5
)2
)1
2
)2
5
(z 2 2 8z 1 16)
[z 2 2 2(
)1
5
(
2]
)2
Checkpoint Factor the polynomial.
5. x 2 1 14x 1 49
6. 9y 2 2 6y 1 1
7. 16x 2 2 40xy 1 25y 2
8. 25r 2 2 20r 2 20
Copyright © Holt McDougal. All rights reserved.
Lesson 9.7 • Algebra 1 Notetaking Guide
241
Your Notes
Example 3
Solve a polynomial equation
1
Solve the equation x 2 1 x 1 } 5 0.
4
1
x2 1 x 1 } 5 0
Write original equation.
4
This equation
has two identical
solutions, because
it has two identical
factors.
50
Multiply each side
by
.
50
Write left side as
a 2 1 2ab 1 b 2.
50
Perfect square
trinomial pattern
50
Zero-product property
x5
Example 4
Solve for x.
Solve a vertical motion problem
Falling Object A brick falls off of a building from a height
of 144 feet. After how many seconds does the brick land
on the ground?
Solution
Use the vertical motion model. The brick fell, so its
initial vertical velocity is
. Find the value of time
t (in seconds) for which the height h (in feet) is
.
h5
Vertical motion model
5
Substitute values.
5
(
)
5
(
)(
Factor out
)
Lesson 9.7 • Algebra 1 Notetaking Guide
Difference of two
squares
or
Zero-product property
or
Solve for t.
The brick lands on the ground
242
.
after it falls.
Copyright © Holt McDougal. All rights reserved.
Your Notes
Checkpoint Solve the equation.
9. m2 2 8m 1 16 5 0
10. w 2 1 16w 1 64 5 0
11. t 2 2 121 5 0
Checkpoint Complete the following exercise.
12. What If? In Example 4, suppose the brick falls from
225
feet. After how many seconds does
a height of }
4
the brick land on the ground?
Homework
Copyright © Holt McDougal. All rights reserved.
Lesson 9.7 • Algebra 1 Notetaking Guide
243
9.8
Factor Polynomials Completely
Goal
Your Notes
p Factor polynomials completely.
VOCABULARY
Factor by grouping
Factor completely
Example 1
Factor out a common binomial
Factor the expression.
a. 3x(x 1 2) 2 2(x 1 2)
b. y 2(y 2 4) 1 3(4 2 y)
Solution
a. 3x(x 1 2) 2 2(x 1 2) 5 (x 1 2)(
)
b. The binomials y 2 4 and 4 2 y are
. Factor
from 4 2 y to obtain a common binomial factor.
y 2(y 2 4) 1 3(4 2 y) 5 y 2(y 2 4)
5 (y 2 4)
Example 2
Factor by grouping
Factor the expression.
a. y 3 1 7y 2 1 2y 1 14
b. y 2 1 2y 1 yx 1 2x
Solution
a. y 3 1 7y 2 1 2y 1 14 5 (
5
5(
Remember that
you can check a
factorization by
multiplying the
factors.
)1(
)1
(
244 Lesson 9.8 • Algebra 1 Notetaking Guide
5(
(
)(
b. y 2 1 2y 1 yx 1 2x 5 (
5
)
)
)1(
)
)1
(
)(
)
(
)
)
Copyright © Holt McDougal. All rights reserved.
Your Notes
Example 3
Factor by grouping
Factor x 3 2 12 1 3x 2 4x 2.
Solution
The terms x3 and 212 have no common factor. Use
the
to rearrange the terms so
that you can group terms with a common factor.
x 3 2 12 1 3x 2 4x 2 5
5
5
5
Checkpoint Factor the expression.
1. 5z(z 2 6) 1 4(z 2 6)
2. 2y 2(y 2 1) 1 7(1 2 y)
3. x 3 2 4x 2 1 5x 2 20
4. n3 1 48 1 6n 1 8n2
GUIDELINES FOR FACTORING POLYNOMIALS
COMPLETELY
To factor a polynomial completely, you should try each
of these steps.
1. Factor out the
common monomial factor.
2. Look for a difference of two squares or a
.
3. Factor a trinomial of the form ax 2 1 bx 1 c into a
product of
factors.
4. Factor a polynomial with four terms by
Copyright © Holt McDougal. All rights reserved.
Lesson 9.8 • Algebra 1 Notetaking Guide
.
245
Your Notes
Factor completely
Example 4
Factor the polynomial completely.
a. x 2 1 3x 2 1
b. 3r 3 2 21r 2 1 30r
c. 9d4 2 4d2
Solution
a. The terms of the polynomial have no common
monomial factor. Also, there are no factors of
that have a sum of
. This polynomial
be factored.
b. 3r 3 2 21r 2 1 30r 5
5
c. 9d4 2 4d2 5
5
Solve a polynomial equation
Example 5
Solve 5x 3 2 25x 2 5 230x.
Solution
5x 3 2 25x 2 5 230x
5x 3 2 25x 2
30x 5 0
30x
to each side.
50
Remember that
you can check
your answers by
substituting each
solution for x in the
original equation.
Factor
out
50
or
x5
246
Write original
equation.
Lesson 9.8 • Algebra 1 Notetaking Guide
Factor
trinomial.
or
x5
.
Zero-product
property
x5
Solve for x.
Copyright © Holt McDougal. All rights reserved.
Your Notes
Example 6
Solve a multi-step problem
Volume A crate in the shape of a rectangular prism has a
volume of 180 cubic feet. The crate has a width of w feet,
a length of (9 2 w) feet, and a height of (w 1 4) feet. The
length is more than half the width. Find the crate's length,
width, and height.
Solution
Step 1 Write and solve an equation for w.
Volume 5
p
p
5
05
05
05
05
05
5 0 or
5 0 or
w5
50
w5
w5
Step 2 Choose the solution that is the correct value for
w. Disregard
, because the width cannot
be
.
You know that the length is more than half the
width. Test the solutions
in the length
expression.
Length 5
Length 5
5
5
or
.
The solution
gives a length of
is more than half the width.
feet, which
Step 3 Find the height.
Height 5
The width is
height is
Copyright © Holt McDougal. All rights reserved.
5
, the length is
5
.
, and the
.
Lesson 9.8 • Algebra 1 Notetaking Guide
247
Your Notes
Checkpoint Factor the polynomial.
5. 22x 3 1 6x 2 1 108x
6. 12y 4 2 75y 2
Checkpoint Complete the following exercises.
7. Solve 2x 3 1 2x 2 5 40x.
Homework
248
8. What If? A box in the shape of a rectangular prism
has a volume of 180 cubic feet. The box has a
length of x feet, a width of (x 1 9) feet, and a height
of (x 2 4) feet. Find the dimensions of the box.
Lesson 9.8 • Algebra 1 Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Words to Review
Give an example of the vocabulary word.
Monomial
Degree of a monomial
Polynomial
Degree of a polynomial
Leading coefficient
Binomial
Trinomial
Roots
Vertical motion model
Perfect square trinomial
Factor by grouping
Factor completely
Review your notes and Chapter 9 by using the
Chapter Review on pages 635–639 of your textbook.
Copyright © Holt McDougal. All rights reserved.
Words to Review • Algebra 1 Notetaking Guide
249