Algebra 1 by Glencoe (Z-Lib - Org) Pages 445 - 472, 480 - 491
Algebra 1 by Glencoe (Z-Lib - Org) Pages 445 - 472, 480 - 491
Algebra 1 by Glencoe (Z-Lib - Org) Pages 445 - 472, 480 - 491
Source: NASA
SCIENTIFIC NOTATION When dealing with very large or very small numbers,
keeping track of place value can be difficult. For this reason, numbers such as these
are often expressed in scientific notation .
Scientific Notation
• Words A number is expressed in scientific notation when it is written as a
product of a factor and a power of 10. The factor must be greater
than or equal to 1 and less than 10.
• Symbols A number in scientific notation is written as a 10n, where 1 a 10
and n is an integer.
Study Tip The following examples show one way of expressing a number that is written
in scientific notation in its decimal or standard notation. Look for a relationship
Reading Math between the power of 10 and the position of the decimal point in the standard
Standard notation is the notation of the number.
way in which you are
1
used to seeing a number 6.59 104 6.59 10,000 4.81 106 4.81 6
written, where the decimal 10
point determines the place 4.81 0.000001
value for each digit of the
number. 65,900 0.00000481
The decimal point moved The decimal point moved
4 places to the right. 6 places to the left.
These examples suggest the following rule for expressing a number written in
scientific notation in standard notation.
Lesson 8-3 Scientific Notation 425
Scientific to Standard Notation
Use these steps to express a number of the form a 10n in standard notation.
1. Determine whether n 0 or n 0.
2. If n 0, move the decimal point in a to the right n places.
If n 0, move the decimal point in a to the left n places.
3. Add zeros, decimal point, and/or commas as needed to indicate place value.
b. 3 105
3 105 0.00003 n 5; move decimal point 5 places to the left.
You will often see large numbers in the media written using a combination of a
number and a word, such as 3.2 million. To write this number in standard notation,
rewrite the word million as 106. The exponent 6 indicates that the decimal point
should be moved 6 places to the right.
Chocolates:
$300,000,000 $3.0 108
All candy: $1,450,000,000 $1.45 109
Application CREDIT CARDS For Exercises 16 and 17, use the following information.
During the year 2000, 1.65 billion credit cards were in use in the United
States. During that same year, $1.54 trillion was charged to these cards.
(Hint: 1 trillion 1 1012 ) Source: U.S. Department of Commerce
16. Express each of these values in standard and then in scientific notation.
17. Find the average amount charged per credit card.
Extra Practice PHYSICS Express the number in each statement in standard notation.
See page 837.
26. There are 2 1011 stars in the Andromeda Galaxy.
27. The center of the moon is 2.389 105 miles away from the center of Earth.
28. The mass of a proton is 1.67265 1027 kilograms.
29. The mass of an electron is 9.1095 1031 kilograms.
56. HAIR GROWTH The usual growth rate of human hair is 3.3 104 meter per
day. If an individual hair grew for 10 years, how long would it be in meters?
(Assume 365 days in a year.)
57. NATIONAL DEBT In April 2001, the national debt was about $5.745 trillion, and
the estimated U.S. population was 283.9 million. About how much was each
U.S. citizen’s share of the national debt at that time?
Online Research Data Update What is the current U.S. population and
amount of national debt? Visit www.algebra1.com/data_update to learn more.
58. BASEBALL The table below lists the greatest yearly salary for a major league
baseball player for selected years.
The contract Alex About how many times as great was the yearly salary of Alex Rodriguez in 2000
Rodriguez signed with as that of George Foster in 1982?
the Texas Rangers on
December 11, 2000,
guarantees him 59. ASTRONOMY The Sun burns about 4.4 106 tons of hydrogen per second.
$25.2 million a year How much hydrogen does the Sun burn in one year? (Hint: First, find the
for 10 seasons. number of seconds in a year and write this number in scientific notation.)
Source: Associated Press
60. CRITICAL THINKING Determine whether each statement is sometimes, always, or
never true. Explain your reasoning.
a. If 1 a 10 and n and p are integers, then (a 10n)p ap 10np.
b. The expression ap 10np in part a is in scientific notation.
www.algebra1.com/self_check_quiz Lesson 8-3 Scientific Notation 429
61. WRITING IN MATH Answer the question that was posed at the beginning of
the lesson.
Why is scientific notation important in astronomy?
Include the following in your answer:
• the mass of each of the planets in standard notation, and
• an explanation of how scientific notation makes presenting and computing
with large numbers easier.
63. SHORT RESPONSE There are an average of 25 billion red blood cells in the
human body and about 270 million hemoglobin molecules in each red blood
cell. Find the average number of hemoglobin molecules in the human body.
Graphing SCIENTIFIC NOTATION You can use a graphing calculator to solve problems
Calculator involving scientific notation. First, put your calculator in scientific mode. To enter
4.5 109, enter 4.5 10 9.
64. (4.5 109)(1.74 102) 65. (7.1 1011)(1.2 105)
66. (4.095 105) (3.15 108) 67. (6 104) (5.5 107)
Determine whether each expression is a monomial. Write yes or no. (Lesson 8-1)
6 v2
71. 3a 4b 72. 73.
n 3
Solve each inequality. Then check your solution and graph it on a number line.
(Lesson 6-1)
74. m 3 17 75. 9 d 9 76. x 11
23
Getting Ready for PREREQUISITE SKILL Evaluate each expression when a 5, b 2, and c 3.
the Next Lesson (To review evaluating expressions, see Lesson 1-2.)
77. 5b2 78. c2 9 79. b3 3ac
80. a2 2a 1 81. 2b4 5b3 b 82. 3.2c3 0.5c2 5.2c
25p10 3 2 4x0y2
5. 4
6k
4.
3 6. 3
5 2
15p 7np (3y z )
Evaluate. Express each result in scientific and standard notation. (Lesson 8-3)
9.2 10 3 3.6 10 7
7. (6.4 103)(7 102) 8. (4 102)(15 106) 9. 5 10. 2
2.3 10 1.2 10
• 2x2 3
2 2
To model this polynomial, you will need 2 blue x x
x2 tiles and 3 red 1 tiles. 1 1 1
• x2 3x 2
2
To model this polynomial, you will need 1 red x x x x
x2 tile, 3 green x tiles, and 2 yellow 1 tiles. 1 1
7. 8.
2
x
2
x
2
x x x x x x
1 1 1 1
9. MAKE A CONJECTURE Write a sentence or two explaining why algebra tiles are
sometimes called area tiles.
Monomial, Binomial,
Expression Polynomial?
or Trinomial?
Yes, 2x 3yz 2x (3yz). The
a. 2x 3yz binomial
expression is the sum of two monomials.
Study Tip b. 8n3 5n2
5
No. 5n2 2 , which is not a monomial. none of these
n
Like Terms c. 8 Yes. 8 is a real number. monomial
Be sure to combine any
Yes. The expression simplifies to
like terms before deciding
if a polynomial is a
d. 4a 2 5a a 9 4a 2 6a 9, so it is the sum of three trinomial
monomial, binomial, or monomials.
trinomial.
TEACHING TIP The degree of a monomial is the sum of the Monomial Degree
exponents of all its variables.
8y 4 4
The degree of a polynomial is the greatest degree
3a 1
of any term in the polynomial. To find the degree of
a polynomial, you must find the degree of each term. 2xy 2z 3 1 2 3 or 6
7 0
Guided Practice State whether each expression is a polynomial. If the expression is a polynomial,
identify it as a monomial, a binomial, or a trinomial.
2z
GUIDED PRACTICE KEY 4. 5x 3xy 2x 5. 6. 9a2 7a 5
5
Find the degree of each polynomial.
7. 1 8. 3x 2 9. 2x2y3 6x4
23. 24.
r
r
x y
54. MULTIPLE BIRTHS The number of quadruplet births Q in the United States
from 1989 to 1998 can be modeled by Q 0.5t3 11.7t2 21.5t 218.6,
where t represents the number of years since 1989. For what values of t does
this model no longer give realistic data? Explain your reasoning.
Multiple Births
From 1980 to 1997, the
number of triplet and PACKAGING For Exercises 55 and 56, use the following
higher births rose 404% r
information.
(from 1377 to 6737 births). A convenience store sells milkshakes in cups with semispherical
This steep climb in multiple lids. The volume of a cylinder is the product of , the square of
births coincides with the
the radius r, and the height h. The volume of a sphere is the h
increased use of fertility 4
drugs. product of , , and the cube of the radius.
3 r
Source: National Center for 55. Write a polynomial that represents the volume of the container.
Health and Statistics
56. If the height of the container is 6 inches and the radius is 2 inches,
find the volume of the container.
www.algebra1.com/self_check_quiz Lesson 8-4 Polynomials 435
57. CRITICAL THINKING Tell whether the following statement is true or false.
Explain your reasoning.
The degree of a binomial can never be zero.
58. WRITING IN MATH Answer the question that was posed at the beginning of
the lesson.
How are polynomials useful in modeling data?
Include the following in your answer:
• a discussion of the accuracy of the equation by evaluating the polynomial for
t {0, 1, 2, 3, 4, 5}, and
• an example of how and why someone might use this equation.
Column A Column B
Getting Ready for PREREQUISITE SKILL Simplify each expression. If not possible, write simplified.
the Next Lesson (To review evaluating expressions, see Lesson 1-5.)
72. 3n 5n 73. 9a2 3a 2a2 74. 12x2 8x 6
75. 3a 5b 4a 7b 76. 4x 3y 6 7x 8 10y
436 Chapter 8 Polynomials
A Preview of Lesson 8-5
2 2 2
x x x x x
3x2 2x 1 →
1
2
3x 2x 1
2
x x x x x
x2 4x 3 → 1 1 1
x2 4x 3
x
2
x
2
x x
1
2 2
x x x x x x
1 1 1
4x 2 2x 2
x x x x x
1 1 1 1
5x 4
Recall that you can subtract a number by adding its additive inverse or opposite.
Similarly, you can subtract a polynomial by adding its opposite.
Activity 3 Use algebra tiles and the additive inverse, or opposite, to find
(5x 4) (2x 3).
To find the difference of 5x 4
5x 4
and 2x 3, add 5x 4 and the
opposite of 2x 3.
5x 4 → x x x x x
1 1 1 1
The opposite of → x x
2x 3 is 2x 3. 1 1 1
2x 3
Method 1 Horizontal
Group like terms together.
Study Tip (3x2 4x 8) (2x 7x2 5)
Adding Columns [3x2 (7x2)] (4x 2x) [8 (5)] Associative and Commutative Properties
When adding like terms in 4x2 2x 3 Add like terms.
column form, remember
that you are adding
Method 2 Vertical
integers. Rewrite each
monomial to eliminate Align the like terms in columns and add.
subtractions. For
example, you could
3x2 4x 8 Notice that terms are in descending order
with like terms aligned.
rewrite 3x2 4x 8 () 7x2 2x 5
as 3x2 (4x) 8.
4x2 2x 3
Method 1 Horizontal
Subtract 7n 4n3 by adding its additive inverse.
Study Tip (3n2 13n3 5n) (7n 4n3)
(3n2 13n3 5n) (7n 4n3) The additive inverse of 7n 4n3 is 7n 4n3.
Inverse of a
Polynomial 3n2 [13n3 (4n3)] [5n (7n)] Group like terms.
When finding the
additive inverse of a 3n2 9n3 2n Add like terms.
polynomial, remember
to find the additive Method 2 Vertical
inverse of every term.
Align like terms in columns and subtract by adding the additive inverse.
3n2 13n3 5n 3n2 13n3 5n
() 4n3 7n Add the opposite. () 4n3 7n
3n2 9n3 2n
Thus, (3n2 13n3 5n) (7n 4n3) 3n2 9n3 2n or, arranged in descending
order, 9n3 3n2 2n.
When polynomials are used to model real-world data, their sums and differences
can have real-world meaning too.
a. Find an equation that models the number of elementary teachers E for this
time period.
Teacher
The educational You can find a model for E by subtracting the polynomial for S from the
requirements for a polynomial for T.
teaching license vary
by state. In 1999, the Total 44n 2216 44n 2216
average public K–12 Secondary () 11n 942 Add the opposite. () 11n 942
teacher salary was
$40,582. Elementary 33n 1274
Esteban Kendra
(5a – 6b) – (2a + 5b) (5a – 6b) – (2a + 5b)
= (–5a + 6b) + (–2a – 5b) = (5a – 6b) + (–2a – 5b)
= –7a + b = 3a – 11b
Application POPULATION For Exercises 10 and 11, use the following information.
From 1990 through 1999, the female population F and the male population M of the
GUIDED PRACTICE KEY United States (in thousands) is modeled by the following equations, where n is the
number of years since 1990. Source: U.S. Census Bureau
F 1247n 126,971 M 1252n 120,741
10. Find an equation that models the total population T in thousands of the United
States for this time period.
11. If this trend continues, what will the population of the United States be in 2010?
NUMBER TRICK For Exercises 34 and 35, use the following information.
Think of a two-digit number whose ones digit is greater than its tens digit. Multiply
the difference of the two digits by 9 and add the result to your original number.
Movies Repeat this process for several other such numbers.
In 1998, attendance 34. What observation can you make about your results?
at movie theaters was
at its highest point in 35. Justify that your observation holds for all such two-digit numbers by letting
40 years with 1.48 billion x equal the tens digit and y equal the ones digit of the original number.
tickets sold for a record (Hint: The original number is then represented by 10x y.)
$6.95 billion in gross
income.
Source: The National Association POSTAL SERVICE For Exercises 36–40, use the
of Theatre Owners length
information below and in the figure at the right.
The U.S. Postal Service restricts the sizes of boxes height
shipped by parcel post. The sum of the length and
the girth of the box must not exceed 108 inches. girth 2(width) 2(height)
width
Suppose you want to make an open box using a 60-by-40 inch piece of cardboard by
cutting squares out of each corner and folding up the flaps. The lid will be made
from another piece of cardboard. You do not know how big the squares should be,
so for now call the length of the side of each square x.
60 in.
x x
x fold x
fold fold 40 in.
x fold x
x x
KEYBOARDING For Exercises 55–59, use the table below that shows the
keyboarding speeds and experience of 12 students. (Lesson 5-2)
Experience
4 7 8 1 6 3 5 2 9 6 7 10
(weeks)
Keyboarding
33 45 46 20 40 30 38 22 52 44 42 55
Speed (wpm)
62. MODEL TRAINS One of the most popular sizes of model trains is called the
1
HO. Every dimension of the HO model measures times that of a real engine.
87
The HO model of a modern diesel locomotive is about 8 inches long. About how
many feet long is the real locomotive? (Lesson 3-6)
Getting Ready for PREREQUISITE SKILL Simplify. (To review the Distributive Property, see Lesson 1-7.)
the Next Lesson 63. 6(3x 8) 64. 2(b 9) 65. 7(5p 4q)
66. 9(3a 5b c) 67. 8(x2 3x 4) 68. 3(2a2 5a 7)
Lesson 8-5 Adding and Subtracting Polynomials 443
Multiplying a Polynomial
by a Monomial
• Find the product of a monomial and a polynomial.
• Solve equations involving polynomials.
Method 2 Vertical
3x2 7x← 10
←
←
When expressions contain like terms, simplify by combining the like terms.
Equation B 20 m 0.06 (300 m) 0.05
20 0.06m 300(0.05) m(0.05) Distributive Property
20 0.06m 15 0.05m Simplify.
The solution is 2.
CHECK y(y 12) y(y 2) 25 2y(y 5) 15 Original equation
2(2 12) 2(2 2) 25 2(2)(2 5) 15 y 2
2(10) 2(4) 25 4(7) 15 Simplify.
20 8 25 28 15 Multiply.
13 13 Add and subtract.
Simplify.
8. t(5t 9) 2t 9. 5n(4n3 6n2 2n 3) 4(n2 7n)
Simplify.
29. d(2d 4) 15d 30. x(4x2 2x) 5x3
31. 3w(6w 4) 2(w2 3w 5) 32. 5n(2n3 n2 8) n(4 n)
33. 10(4m3 3m 2) 2m(3m2 7m 1)
34. 4y(y2 8y 6) 3(2y3 5y2 2)
35. 3c2(2c 7) 4c(3c2 c 5) 2(c2 4)
36. 4x2(x 2) 3x(5x2 2x 6) 5(3x2 4x)
446 Chapter 8 Polynomials
GEOMETRY Find the area of each shaded region in simplest form.
37. 4x 38. 5p
3x 6
3x 2 2x 2p 1 3p 4
52. CLASS TRIP Mr. Smith’s American History class will take taxis from their hotel
in Washington, D.C., to the Lincoln Memorial. The fare is $2.75 for the first mile
and $1.25 for each additional mile. If the distance is m miles and t taxis are
needed, write an expression for the cost to transport the group.
x 2.5
Start
If the radius of the inside lane is x and each lane is 2.5 feet wide, how
far apart should the officials start the runners in the two inside lanes?
(Hint: Circumference of a circle: C 2r, where r is the radius of the circle)
63. WRITING IN MATH Answer the question that was posed at the beginning of
the lesson.
How is finding the product of a monomial and a polynomial related to
finding the area of a rectangle?
Include the following in your answer:
• the product of 2x and x 3 derived algebraically, and
• a representation of another product of a monomial and a polynomial using
algebra tiles and multiplication.
65. A plumber charges $70 for the first thirty minutes of each house call plus $4 for
each additional minute that she works. The plumber charges Ke-Min $122 for
her time. What amount of time, in minutes, did the plumber work?
A 43 B 48 C 58 D 64
448 Chapter 8 Polynomials
Maintain Your Skills
Mixed Review Find each sum or difference. (Lesson 8-5)
66. (4x2 5x) (7x2 x) 67. (3y2 5y 6) (7y2 9)
68. (5b 7ab 8a) (5ab 4a) 69. (6p3 3p2 7) (p3 6p2 2p)
Define a variable, write an inequality, and solve each problem. Then check your
solution. (Lesson 6-3)
74. Six increased by ten times a number is less than nine times the number.
75. Nine times a number increased by four is no less than seven decreased by
thirteen times the number.
Write an equation of the line that passes through each pair of points. (Lesson 5-4)
76. (3, 8), (1, 4) 77. (4, 5), (2, 7) 78. (3, 1), (3, 2)
79. EXPENSES Kristen spent one fifth of her money on gasoline to fill up her car.
Then she spent half of what was left for a haircut. She bought lunch for $7.
When she got home, she had $13 left. How much money did Kristen have
originally? (Lesson 3-4)
For Exercises 80 and 81, use each set of data to make a stem-and-leaf plot.
(Lesson 2-5)
80. 49 51 55 62 47 32 56 57 48 47 33 68 53 45 30
81. 21 18 34 30 20 15 14 10 22 21 18 43 44 20 18
Getting Ready for PREREQUISITE SKILL Simplify. (To review products of polynomials, see Lesson 8-1.)
the Next Lesson 82. (a)(a) 83. 2x(3x2)
84. 3y2(8y2) 85. 4y(3y) 4y(6)
86. 5n(2n2) (5n)(8n) (5n)(4) 87. 3p2(6p2) 3p2(8p) 3p2(12)
Arrange the terms of each polynomial so that the powers of x are in ascending order. (Lesson 8-4)
5. 4x2 9x 12 5x3 6. 2xy4 x3y5 5x5y 13x2
Multiplying Polynomials
You can use algebra tiles to find the product of two binomials.
x 1 1 1 1 1 x5
2
x x x x x x x
x2
1 x 1 1 1 1 1
1 x 1 1 1 1 1
The rectangle consists of 1 blue x2 tile, 7 green x tiles, and 10 yellow 1 tiles. The area
of the rectangle is x2 7x 10. Therefore, (x 2)(x 5) x2 7x 10.
x 1 1 1 1 x4
2
x x x x x x
x1
1 x
The rectangle will have a width of x 3 and a length of 2x 1. Mark off the
dimensions on a product mat. Then begin to make the rectangle with algebra tiles.
x x 1 2x 1
2 2
x x x x
x3
1 x
1 x
1 x
24 24 24
36 36 36
144 144 144
720 720
6 24 6(20 4) 864
120 24 or 144 30 24 30(20 4)
600 120 or 720
Method 2 Horizontal
(x 3)(x 2) x(x 2) 3(x 2) Distributive Property
An alternative method for finding the product of two binomials can be shown
using algebra tiles.
452 Chapter 8 Polynomials
Example 3 Apply the Sum of a Square
GENETICS The Punnett square shows the
possible gene combinations of a cross between
two pea plants. Each plant passes along one Tt
dominant gene T for tallness and one
T t
recessive gene t for shortness.
More About . . .
Show how combinations can be modeled by TT Tt
the square of a binomial. Then determine T
pure tall hybrid tall
what percent of the offspring will be pure
tall, hybrid tall, and pure short. Tt
Each parent has half the genes necessary for
t Tt tt
tallness and half the genes necessary for
shortness. The makeup of each parent can hybrid tall pure short
be modeled by 0.5T 0.5t. Their offspring
can be modeled by the product of 0.5T 0.5t
Geneticist and 0.5T 0.5t or (0.5T 0.5t)2.
Laboratory geneticists work If we expand this product, we can determine the possible heights of the offspring.
in medicine to find cures
for disease, in agriculture
(a b)2 a2 2ab b2 Square of a Sum
to breed new crops and (0.5T 0.5t)2 (0.5T)2 2(0.5T)(0.5t) (0.5t)2 a 0.5T and b 0.5t
livestock, and in police
0.25T2 0.5Tt 0.25t2 Simplify.
work to identify criminals.
0.25TT 0.5Tt 0.25tt T2 TT and t2 tt
Online Research
Thus, 25% of the offspring are TT or pure tall, 50% are Tt or hybrid tall, and 25%
For information about
are tt or pure short.
a career as a
geneticist, visit:
www.algebra1.com/ PRODUCT OF A SUM AND A DIFFERENCE You can use the diagram
careers below to find the pattern for the product of a sum and a difference of the same two
terms, (a b)(a b). Recall that a b can be rewritten as a (b).
a a
2
ab
a
2
ab ab b
2
ab
2
b ab b
a
2
b 2
a2 (b2)
a2 b2
The resulting product, a2 b2, has a special name. It is called a difference of
squares . Notice that this product has no middle term.
The following list summarizes the special products you have studied.
Special Products
• Square of a Sum (a b)2 a2 2ab b2
• Square of a Difference (a b)2 a2 2ab b2
• Product of a Sum and a Difference (a b)(a b) a2 b2
Concept Check 1. Compare and contrast the pattern for the square of a sum with the pattern for
the square of a difference.
2. Explain how the square of a difference and the difference of squares differ.
3. Draw a diagram to show how you would use algebra tiles to model the product
of x 3 and x 3, or (x 3)2.
4. OPEN ENDED Write two binomials whose product is a difference of squares.
Cinnamon
Extending 51. Does a pattern exist for the cube of a sum, (a b)3?
the Lesson a. Investigate this question by finding the product of (a b)(a b)(a b).
b. Use the pattern you discovered in part a to find (x 2)3.
c. Draw a diagram of a geometric model for the cube of a sum.
Use elimination to solve each system of equations. (Lessons 7-3 and 7-4)
3 1
62. x y 5 63. 2x y 10 64. 2x 4 3y
4 5
3 1 5x 3y 3 3y x 11
x y 5
4 5
Write the slope-intercept form of an equation that passes through the given point
and is perpendicular to the graph of each equation. (Lesson 5-6)
65. 5x 5y 35, (3, 2) 66. 2x 5y 3, (2, 7) 67. 5x y 2, (0, 6)
Find the nth term of each arithmetic sequence described. (Lesson 4-7)
68. a1 3, d 4, n 18 69. 5, 1, 7, 13, … for n 12
Choose a term from the vocabulary list that best matches each example.
1
1. 43 3 2. (n3)5 n15
4
4x2y x
3. 3 2 4. 4x2
8xy 2y
5. x2 3x 1 6. 20 1
7. x4 3x3 2x2 1 8. (x 3)(x 4) x2 4x 3x 12
9. x2 2 10. (a3 b)(2ab2) 2a4 b 3
2 Simplify (2x2y3)3.
(2x2y3)3 23(x2)3(y3)3 Power of a Product
8x6y9 Power of a Power
2x6y
Example Simplify 22 . Assume that x and y are not equal to zero.
8x y
2x6y 2 x6 y
8x y
22
8 x2 y2
Group the powers with the same base.
1
(x6 2)(y1 2)
4
Quotient of Powers
x4
Simplify.
4y
27b2 3
(3a bc ) 2 2 16a3b2x4y
23. 3 24. 2 25.
14b 3 4
18a b c 48a bxy
4 3
5 8 1 2 5xy2
(a) b (4a ) 0
26. 5 2 27. 4 2 28.
2 6
ab (2a ) 35x y
Exercises Express each number in standard notation. See Example 1 on page 426.
8-4 Polynomials
See pages Concept Summary
432–436.
• A polynomial is a monomial or a sum of monomials.
• A binomial is the sum of two monomials, and a trinomial is the sum of
three monomials.
• The degree of a monomial is the sum of the exponents of all its variables.
• The degree of the polynomial is the greatest degree of any term. To find
the degree of a polynomial, you must find the degree of each term.
Exercises Find the degree of each polynomial. See Example 3 on page 433.
38. n 2p2 39. 29n2 17n2t2 40. 4xy 9x3z2 17rs3
41. 6x5y 2y4 4 8y2 42. 3ab3 5a2b2 4ab 43. 19m3n4 21m5n
Arrange the terms of each polynomial so that the powers of x are in descending
order. See Example 5 on page 433.
44. 3x4 x x2 5 45. 2x2y3 27 4x4 xy 5x3y2
Exercises Find each sum or difference. See Examples 1 and 2 on pages 439 and 440.
46. (2x2 5x 7) (3x3 2)
x2 47. (x2 6xy 7y2) (3x2 xy y2)
48. (7z2 4) (3z2 2z 6) 49. (13m4 7m 10) (8m4 3m 9)
50. (11m2n2 4mn 6) (5m2n2 6mn 17)
51. (5p2 3p 49) (2p2 5p 24)
O 3x 4x 4
2 Combine like terms.
r2 10r 25 Simplify.
Exercises Find each product. See Examples 1, 2, and 4 on pages 459 and 461.
64. (x 6)(x 6) 65. (4x 7)2 66. (8x 5)2
67. (5x 3y)(5x 3y) 68. (6a 5b)2 69. (3m 4n)2
Find the degree of each polynomial. Then arrange the terms so that the
powers of y are in descending order.
20. 2y2 8y4 9y 21. 5xy 7 2y4 x2y3
Simplify. 5x 2 13x 24
A
1
y x 1
Test-Taking Tip
5 x Question 5 When you write an equation, check
B y 5x 1 O
that the given values make a true statement. For
C
1
y x 5 example, in Question 5, substitute the values of the
5 coordinates (1, 4) into your equation to check.
D y 5x 5
470 Chapter 8 Polynomials
Aligned and
verified by
6(x 1)
Part 2 Short Response/Grid In 18. 4x 10 20 3
8
13. Find the y-intercept of the line represented 21. 5.01 102 50.1 104
by 3x 2y 8 0. (Lesson 5-4) (Lesson 8-3)
14. Graph the solution of the linear inequality 22. the degree of the degree of
3x y 2. (Lesson 6-6) x2 5 6x 13x3 10 y 2y2 4y3
15. Let P 3x2 2x 1 and Q x2 2x 2. (Lesson 8-4)
Find P Q. (Lesson 8-5)
23. m2 n2 10 and mn 6
16. Find (x2 1)(x 3). (Lesson 8-7) (m n)2 (m n)2
(Lesson 8-8)
Part 3 Quantitative Comparison
Compare the quantity in Column A and the Part 4 Open Ended
quantity in Column B. Then determine
whether: Record your answers on a sheet of paper.
Show your work.
A the quantity in Column A is greater,
24. Use the rectangular prism below to solve
B the quantity in Column B is greater,
the following problems. (Lessons 8-1 and 8-7)
C the two quantities are equal, or
D the relationship cannot be determined
m4
from the information given.