1. Warm Up
Evaluate each expression for the given values of the variables.
1. x3y2 for x = –1 and y = 10
for x = 4 and y = (–7)
Write each number as a power of the given base.
–100
3. 64; base 4 43
4. –27; base (–3)
(–3)3

2. Objectives Evaluate expressions containing zero and integer exponents.
Simplify expressions containing zero and integer exponents.

3. You have seen positive exponents
You have seen positive exponents. Recall that to simplify 32, use 3 as a factor 2 times: 32 = 3  3 = 9.
But what does it mean for an exponent to be negative or 0? You can use a table and look for a pattern to figure it out.
Power
Value 55 54 53 52 51 50
5–1
5–2
3125
625
125 25 5
 5
 5
 5
 5

4. When the exponent decreases by one, the value of the power is divided by 5. Continue the pattern of dividing by 5.

5. Base x
Exponent
Remember! 4

7. Notice the phrase “nonzero number” in the previous table
Notice the phrase “nonzero number” in the previous table. This is because 00 and 0 raised to a negative power are both undefined. For example, if you use the pattern given above the table with a base of 0 instead of 5, you would get 0º = . Also 0–6 would be = . Since division by 0 is undefined, neither value exists.

8. 2–4 is read “2 to the negative fourth power.”
Reading Math

9. Example 1: Application
One cup is 2–4 gallons. Simplify this expression.
cup is equal to

10. Check It Out! Example 1
A sand fly may have a wingspan up to 5–3 m. Simplify this expression.
5-3 m is equal to

11. Example 2: Zero and Negative Exponents
Simplify.
A. 4–3
B. 70
Any nonzero number raised to the zero power is 1.
7º = 1
C. (–5)–4
D. –5–4

12. In (–3)–4, the base is negative because the negative sign is inside the parentheses. In –3–4 the base (3) is positive.
Caution

13. Check It Out! Example 2
Simplify.
a. 10–4
b. (–2)–4
c. (–2)–5
d. –2–5

14. Example 3A: Evaluating Expressions with Zero and Negative Exponents
Evaluate the expression for the given value of the variables.
x–2 for x = 4
Substitute 4 for x.
Use the definition

15. Example 3B: Evaluating Expressions with Zero and Negative Exponents
Evaluate the expression for the given values of the variables.
–2a0b-4 for a = 5 and b = –3
Substitute 5 for a and –3 for b.
Evaluate expressions with exponents.
Write the power in the denominator as a product.
Evaluate the powers in the product.
Simplify.

16. Check It Out! Example 3a
Evaluate the expression for the given value of the variable.
p–3 for p = 4
Substitute 4 for p.
Evaluate exponent.
Write the power in the denominator as a product.
Evaluate the powers in the product.

17. Check It Out! Example 3b
Evaluate the expression for the given values of the variables.
for a = –2 and b = 6
Substitute –2 for a and 6 for b.
Evaluate expressions with exponents.
Write the power in the denominator as a product.
Evaluate the powers in the product. 2
Simplify.

18. What if you have an expression with a negative exponent in a denominator, such as ?
Definition of a negative exponent.
Substitute –8 for n.
Simplify the exponent on the right side.
An expression that contains negative or zero exponents is not considered to be simplified. Expressions should be rewritten with only positive exponents.
So if a base with a negative exponent is in a denominator, it is equivalent to the same base with the opposite (positive) exponent in the numerator.

19. Example 4: Simplifying Expressions with Zero and Negative Numbers
A. 7w–4

20. Example 4: Simplifying Expressions with Zero and Negative NumbersC.
and

21. Check It Out! Example 4
Simplify.
a. 2r0m–3
rº = 1 and b. c.

22. Lesson Quiz: Part I
1. A square foot is 3–2 square yards. Simplify this expression.
Simplify.
2. 2–6
3. (–7)–3
4. 60 1
5. –112
–121

23. Lesson Quiz: Part II
Evaluate each expression for the given value(s) of the variables(s). 6.
x–4 for x =10 7.
for a = 6 and b = 3

24. Warm Up
Evaluate each expression.
 1,000
 1,000
 100
 100
5. 104
6. 10–4
123,000
0.123
0.3
10,000
0.0001 1

25. Objectives Evaluate and multiply by powers of 10.
Convert between standard notation and scientific notation.

26. Vocabulary
scientific notation

27. The table shows relationships between several powers of 10.
Each time you divide by 10, the exponent decreases by 1 and the decimal point moves one place to the left.

28. The table shows relationships between several powers of 10.
Each time you multiply by 10, the exponent increases by 1 and the decimal point moves one place to the right.

30. Example 1: Evaluating Powers of 10
Find the value of each power of 10.
A. 10–6
B. 104
C. 109
Start with 1 and move the decimal point six places to the left.
Start with 1 and move the decimal point four places to the right.
Start with 1 and move the decimal point nine places to the right.
10,000
1,000,000,000

31. You may need to add zeros to the right or left of a number in order to move the decimal point in that direction.
Writing Math

32. Check It Out! Example 1
Find the value of each power of 10.
a. 10–2
b. 105
c. 1010
Start with 1 and move the decimal point two places to the left.
Start with 1 and move the decimal point five places to the right.
Start with 1 and move the decimal point ten places to the right.
0.01
100,000
10,000,000,000

33. If you do not see a decimal point in a number, it is understood to be at the end of the number.
Reading Math

34. Example 2: Writing Powers of 10
Write each number as a power of 10.
A. 1,000,000 B
C. 1,000
The decimal point is six places to the right of 1, so the exponent is 6.
The decimal point is four places to the left of 1, so the exponent is –4.
The decimal point is three places to the right of 1, so the exponent is 3.

35. Check It Out! Example 2
Write each number as a power of 10.
a. 100,000,000 b
c. 0.1
The decimal point is eight places to the right of 1, so the exponent is 8.
The decimal point is four places to the left of 1, so the exponent is –4.
The decimal point is one place to the left of 1, so the exponent is –1.

36. You can also move the decimal point to find the value of any number multiplied by a power of 10. You start with the number rather than starting with 1.
Multiplying by Powers of 10

37. Example 3: Multiplying by Powers of 10
Find the value of each expression.
A  108
Move the decimal point 8 places to the right.
2,389,000,000
B. 467  10–3
4 6 7
Move the decimal point 3 places to the left.
0.467

38. Check It Out! Example 3
Find the value of each expression.
a  105
Move the decimal point 5 places to the right.
85,340,000
b  10–2
Move the decimal point 2 places to the left.

39. Scientific notation is a method of writing numbers that are very large or very small. A number written in scientific notation has two parts that are multiplied.
The first part is a number that is greater than or equal to 1 and less than 10.
The second part is a power of 10.

40. Example 4A: Astronomy Application
Saturn has a diameter of about km. Its distance from the Sun is about 1,427,000,000 km.
Write Saturn’s diameter in standard form.
Move the decimal point 5 places to the right.
120,000 km

41. Example 4B: Astronomy Application
Saturn has a diameter of about km. Its distance from the Sun is about 1,427,000,000 km.
Write Saturn’s distance from the Sun in scientific notation.
Count the number of places you need to move the decimal point to get a number between 1 and 10.
1,427,000,000
1,4 2 7,0 0 0,0 0 0
9 places
Use that number as the exponent of 10.
1.427  109 km

42. Standard form refers to the usual way that numbers are written—not in scientific notation.
Reading Math

43. Check It Out! Example 4a
Use the information above to write Jupiter’s diameter in scientific notation.
143,000 km
Count the number of places you need to move the decimal point to get a number between 1 and 10.
5 places
Use that number as the exponent of 10.
1.43  105 km

44. Check It Out! Example 4b
Use the information above to write Jupiter’s orbital speed in standard form.
Move the decimal point 4 places to the right.
13,000 m/s

45. Example 5: Comparing and Ordering Numbers in Scientific Notation
Order the list of numbers from least to greatest.
Step 1 List the numbers in order by powers of 10.
Step 2 Order the numbers that have the same power of 10

46. Check It Out! Example 5
Order the list of numbers from least to greatest.
Step 1 List the numbers in order by powers of 10.
2  10-12, 4  10-3, 5.2  10-3, 3  1014, 4.5  1014, 4.5  1030
Step 2 Order the numbers that have the same power of 10

47. Lesson Quiz: Part I
Find the value of each expression. 1. 2.
3. The Pacific Ocean has an area of about 6.4 х 107 square miles. Its volume is about 170,000,000 cubic miles.
a. Write the area of the Pacific Ocean in standard
3,745,000
form.
b. Write the volume of the Pacific Ocean in scientific notation.
1.7  108 mi3

48. Lesson Quiz: Part II
Find the value of each expression.
4. Order the list of numbers from least to greatest

49. Multiplication Properties of Exponents 7-3
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 1

50. Warm Up Write each expression using an exponent. 1. 2 • 2 • 2
2. x • x • x • x 3.
Write each expression without using an exponent.
4. 43
5. y2
6. m–4 23
4 • 4 • 4
y • y

51. Objective
Use multiplication properties of exponents to evaluate and simplify expressions.

52. You have seen that exponential expressions are useful when writing very small or very large numbers. To perform operations on these numbers, you can use properties of exponents. You can also use these properties to simplify your answer.
In this lesson, you will learn some properties that will help you simplify exponential expressions containing multiplication.

54. Products of powers with the same base can be found by writing each power as a repeated multiplication.
Notice the relationship between the exponents in the factors and the exponents in the product
5 + 2 = 7.

56. Example 1: Finding Products of Powers
Simplify. A.
Since the powers have the same base, keep the base and add the exponents. B.
Group powers with the same base together.
Add the exponents of powers with the same base.

57. Example 1: Finding Products of Powers
Simplify. C.
Group powers with the same base together.
Add the exponents of powers with the same base. D.
Group the positive exponents and add since they have the same base 1
Add the like bases.

58. A number or variable written without an exponent actually has an exponent of 1.
Remember!
10 = 101
y = y1

59. Check It Out! Example 1
Simplify. a.
Since the powers have the same base, keep the base and add the exponents. b.
Group powers with the same base together.
Add the exponents of powers with the same base.

60. Check It Out! Example 1
Simplify. c.
Group powers with the same base together.
Add.

61. Check It Out! Example 1
Simplify. d.
Group the first two and second two terms.
Divide the first group and add the second group.
Multiply.

62. Example 2: Astronomy Application
Light from the Sun travels at about miles per second. It takes about 15,000 seconds for the light to reach Neptune. Find the approximate distance from the Sun to Neptune. Write your answer in scientific notation.
distance = rate  time
Write 15,000 in scientific notation.
Use the Commutative and Associative Properties to group.
Multiply within each group. mi

63. Check It Out! Example 2
Light travels at about miles per second. Find the approximate distance that light travels in one hour. Write your answer in scientific notation.
distance = rate  time
Write 3,600 in scientific notation.
Use the Commutative and Associative Properties to group.
Multiply within each group.

64. To find a power of a power, you can use the meaning of exponents.
Notice the relationship between the exponents in the original power and the exponent in the final power:

66. Example 3: Finding Powers of Powers
Simplify.
Use the Power of a Power Property.
Simplify.
Use the Power of a Power Property.
Zero multiplied by any number is zero 1
Any number raised to the zero power is 1.

67. Example 3: Finding Powers of Powers
Simplify. C.
Use the Power of a Power Property.
Simplify the exponent of the first term.
Since the powers have the same base, add the exponents.
Write with a positive exponent.

68. Check It Out! Example 3
Simplify.
Use the Power of a Power Property.
Simplify.
Use the Power of a Power Property.
Zero multiplied by any number is zero.
Any number raised to the zero power is 1. 1

69. Check It Out! Example 3c
Simplify.
Use the Power of a Power Property. c.
Simplify the exponents of the two terms.
Since the powers have the same base, add the exponents.

70. Powers of products can be found by using the meaning of an exponent.

71. Example 4: Finding Powers of Products
Simplify. A.
Use the Power of a Product Property.
Simplify. B.
Use the Power of a Product Property.
Simplify.

72. Example 4: Finding Powers of Products
Simplify. C.
Use the Power of a Product Property.
Use the Power of a Product Property.
Simplify.

73. Check It Out! Example 4
Simplify.
Use the Power of a Product Property.
Simplify.
Use the Power of a Product Property.
Use the Power of a Product Property.
Simplify.

74. Check It Out! Example 4
Simplify. c.
Use the Power of a Product Property.
Use the Power of a Product Property.
Combine like terms.
Write with a positive exponent.

75. Lesson Quiz: Part I
Simplify.
1. 32• 34 3. 5. 7. 2.
(x3)2 4. 6.

76. Lesson Quiz: Part II
7. The islands of Samoa have an approximate area of 2.9  103 square kilometers. The area of Texas is about 2.3  102 times as great as that of the islands. What is the approximate area of Texas? Write your answer in scientific notation.

77. Division Properties of Exponents
7-4
Division Properties of Exponents
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 1

78. Warm Up
Simplify.
1. (x2)3 3. 5. 2. 4. 6.
Write in Scientific Notation. 7. 8.

79. Objective
Use division properties of exponents to evaluate and simplify expressions.

80. A quotient of powers with the same base can be found by writing the powers in a factored form and dividing out common factors.
Notice the relationship between the exponents in the original quotient and the exponent in the final answer: 5 – 3 = 2.

82. Example 1: Finding Quotients of Powers
Simplify. A. B.

83. Example 1: Finding Quotients of Powers
Simplify. C. D.

84. Both and 729 are considered to be simplified.
Helpful Hint

85. Check It Out! Example 1
Simplify. a. b.

86. Check It Out! Example 1
Simplify. c. d.

87. Example 2: Dividing Numbers in Scientific Notation
Simplify and write the answer in scientific notation
Write as a product of quotients.
Simplify each quotient.
Simplify the exponent.
Write 0.5 in scientific notation as 5 x
The second two terms have the same base, so add the exponents.
Simplify the exponent.

88. You can “split up” a quotient of products into a product of quotients:
Example:
Writing Math

89. Check It Out! Example 2
Simplify and write the answer in scientific notation.
Write as a product of quotients.
Simplify each quotient.
Simplify the exponent.
Write 1.1 in scientific notation as 11 x
The second two terms have the same base, so add the exponents.
Simplify the exponent.

90. Example 3: Application
The Colorado Department of Education spent about dollars in fiscal year on public schools. There were about students enrolled in public school. What was the average spending per student? Write your answer in standard form.
To find the average spending per student, divide the total debt by the number of students.
Write as a product of quotients.

91. Example 3 Continued
The Colorado Department of Education spent about dollars in fiscal year on public schools. There were about students enrolled in public school. What was the average spending per student? Write your answer in standard form.
To find the average spending per student, divide the total debt by the number of students.
Simplify each quotient.
Simplify the exponent.
Write in standard form.
The average spending per student is $5,800.

92. Check It Out! Example 3
In 1990, the United States public debt was about dollars. The population of the United States was about people. What was the average debt per person? Write your answer in standard form.
To find the average debt per person, divide the total debt by the number of people.
Write as a product of quotients.

93. Check It Out! Example 3 Continued
In 1990, the United States public debt was about dollars. The population of the United States was about people. What was the average debt per person? Write your answer in standard form.
To find the average debt per person, divide the total debt by the number of people.
Simplify each quotient.
Simplify the exponent.
Write in standard form.
The average debt per person was $12,800.

94. A power of a quotient can be found by first writing the numerator and denominator as powers.
Notice that the exponents in the final answer are the same as the exponent in the original expression.

96. Example 4A: Finding Positive Powers of Quotient
Simplify.
Use the Power of a Quotient Property.
Simplify.

97. Example 4B: Finding Positive Powers of Quotient
Simplify.
Use the Power of a Product Property.
Use the Power of a Product Property:
Simplify and use the Power of a Power Property:

98. Example 4C: Finding Positive Powers of Quotient
Simplify.
Use the Power of a Product Property.
Use the Power of a Product Property:
Use the Power of a Product Property:

99. Example 4C Continued
Simplify.
Use the Power of a Product Property:

100. Check It Out! Example 4a
Simplify.
Use the Power of a Quotient Property.
Simplify.

101. Check It Out! Example 4b
Simplify.

102. Check It Out! Example 4c
Simplify.

103. . Remember that What if x is a fraction?
Write the fraction as division.
Use the Power of a Quotient Property.
Multiply by the reciprocal.
Simplify.
Use the Power of a Quotient Property.
Therefore,

105. Example 5A: Finding Negative Powers of Quotients
Simplify.
Rewrite with a positive exponent.
Use the Powers of a Quotient Property .
and

106. Example 5B: Finding Negative Powers of Quotients
Simplify.

107. Example 5C: Finding Negative Powers of Quotients
Simplify.
Rewrite each fraction with a positive exponent.
Use the Power of a Quotient Property.
Use the Power of a Product Property:
(3)2 (2n)3 = 32  23n3
and (2)2  (6m)3 = 22  63m3

108. Example 5C: Finding Negative Powers of Quotients
Simplify.
Square and cube terms. 1 24 2 12
Divide out common factors.
Simplify.

109. Whenever all of the factors in the numerator or the denominator divide out, replace them with 1.
Helpful Hint

110. Check It Out! Example 5a
Simplify.
Rewrite with a positive exponent.
Use the power of a Quotient Property.
93=729 and 43 = 64.

111. Check It Out! Example 5b
Simplify.
Rewrite with a positive exponent.
Use the Power of a Quotient Property.
Use the Power of a Power Property: (b2c3)4= b2•4c3•4 = b8c12 and (2a)4= 24a4= 16a4.

112. Check It Out! Example 5c
Simplify.
Rewrite each fraction with a positive exponent.
Use the Power of a Quotient Property.
Use the Power of a Product Property: (3)2= 9.
Add exponents and divide out common terms.

113. 1. 5.
Lesson Quiz: Part I
Simplify. 2.

114. Lesson Quiz: Part II
Simplify.
6. Simplify (3  1012) ÷ (5  105) and write the answer in scientific notation.
6  106
7. The Republic of Botswana has an area of 6  105 square kilometers. Its population is about 1.62  106. What is the population density of Botswana? Write your answer in standard form.
2.7 people/km2

115. Warm Up Evaluate each expression for the given value of x.
1. 2x + 3; x = x2 + 4; x = –3
3. –4x – 2; x = –1 4. 7x2 + 2x = 3
Identify the coefficient in each term.
5. 4x y3
7. 2n –54 7 13 2 69 4 1 2 –1

116. Objectives
Classify polynomials and write polynomials in standard form.
Evaluate polynomial expressions.

117. Vocabulary monomial degree of a monomial polynomial
degree of a polynomial
standard form of a
leading coefficient
quadratic
cubic
binomial
trinomial

118. A monomial is a number, a variable, or a product of numbers and variables with whole-number exponents.
The degree of a monomial is the sum of the exponents of the variables. A constant has degree 0.

119. Example 1: Finding the Degree of a Monomial
Find the degree of each monomial. A.
4p4q3
The degree is 7.
Add the exponents of the variables: = 7.
B. 7ed
The degree is 2.
Add the exponents of the variables: 1+ 1 = 2.
C. 3
The degree is 0.
Add the exponents of the variables: 0 = 0.

120. The terms of an expression are the parts being added or subtracted
The terms of an expression are the parts being added or subtracted. See Lesson 1-7.
Remember!

121. Check It Out! Example 1
Find the degree of each monomial. a.
1.5k2m
The degree is 3.
Add the exponents of the variables: = 3. b. 4x
The degree is 1.
Add the exponents of the variables: 1 = 1. b.
2c3
The degree is 3.
Add the exponents of the variables: 3 = 3.

122. A polynomial is a monomial or a sum or difference of monomials.
The degree of a polynomial is the degree of the term with the greatest degree.

123. Example 2: Finding the Degree of a Polynomial
Find the degree of each polynomial.
A. 11x7 + 3x3
11x7: degree 7 3x3: degree 3
Find the degree of each term.
The degree of the polynomial is the greatest degree, 7. B.
:degree 3
:degree 4
–5: degree 0
Find the degree of each term.
The degree of the polynomial is the greatest degree, 4.

124. Check It Out! Example 2
Find the degree of each polynomial.
a. 5x – 6
5x: degree 1
–6: degree 0
Find the degree of each term.
The degree of the polynomial is the greatest degree, 1.
b. x3y2 + x2y3 – x4 + 2
Find the degree of each term.
x3y2: degree 5
x2y3: degree 5
–x4: degree 4
2: degree 0
The degree of the polynomial is the greatest degree, 5.

125. The terms of a polynomial may be written in any order
The terms of a polynomial may be written in any order. However, polynomials that contain only one variable are usually written in standard form.
The standard form of a polynomial that contains one variable is written with the terms in order from greatest degree to least degree. When written in standard form, the coefficient of the first term is called the leading coefficient.

126. Example 3A: Writing Polynomials in Standard Form
Write the polynomial in standard form. Then give the leading coefficient.
6x – 7x5 + 4x2 + 9
Find the degree of each term. Then arrange them in descending order:
6x – 7x5 + 4x2 + 9
–7x5 + 4x2 + 6x + 9
Degree 1 5 2
–7x5 + 4x2 + 6x + 9.
The standard form is
The leading
coefficient is –7.

127. Example 3B: Writing Polynomials in Standard Form
Write the polynomial in standard form. Then give the leading coefficient.
y2 + y6 − 3y
Find the degree of each term. Then arrange them in descending order:
y2 + y6 – 3y
y6 + y2 – 3y
Degree 2 6 1
The standard form is
The leading
coefficient is 1.
y6 + y2 – 3y.

128. A variable written without a coefficient has a coefficient of 1.
Remember!
y5 = 1y5

129. Check It Out! Example 3a
Write the polynomial in standard form. Then give the leading coefficient.
16 – 4x2 + x5 + 9x3
Find the degree of each term. Then arrange them in descending order:
16 – 4x2 + x5 + 9x3
x5 + 9x3 – 4x2 + 16
Degree 2 5 3
The standard form is
The leading
coefficient is 1.
x5 + 9x3 – 4x

130. Check It Out! Example 3b
Write the polynomial in standard form. Then give the leading coefficient.
18y5 – 3y8 + 14y
Find the degree of each term. Then arrange them in descending order:
18y5 – 3y8 + 14y
–3y8 + 18y5 + 14y
Degree 5 8 1
The standard form is
The leading
coefficient is –3.
–3y8 + 18y5 + 14y.

131. Some polynomials have special names based on their degree and the number of terms they have.
Monomial
Binomial
Trinomial
Polynomial
4 or more 1 2 3 1 2
Constant
Linear
Quadratic 3 4 5
6 or more
6th,7th,degree and so on
Cubic
Quartic
Quintic

132. Example 4: Classifying Polynomials
Classify each polynomial according to its degree and number of terms.
A. 5n3 + 4n
5n3 + 4n is a cubic binomial.
Degree 3 Terms 2
B. 4y6 – 5y3 + 2y – 9
4y6 – 5y3 + 2y – 9 is a
6th-degree polynomial.
Degree 6 Terms 4
C. –2x
–2x is a linear monomial.
Degree 1 Terms 1

133. Check It Out! Example 4
Classify each polynomial according to its degree and number of terms.
a. x3 + x2 – x + 2
x3 + x2 – x + 2 is a cubic polymial.
Degree 3 Terms 4
b. 6
6 is a constant monomial.
Degree 0 Terms 1
–3y8 + 18y5 + 14y is an 8th-degree trinomial.
c. –3y8 + 18y5 + 14y
Degree 8 Terms 3

134. Example 5: Application
A tourist accidentally drops her lip balm off the Golden Gate Bridge. The bridge is 220 feet from the water of the bay. The height of the lip balm is given by the polynomial –16t , where t is time in seconds. How far above the water will the lip balm be after 3 seconds?
Substitute the time for t to find the lip balm’s height.
–16t
–16(3)
The time is 3 seconds.
–16(9) + 200
Evaluate the polynomial by using the order of operations. – 76

135. Example 5: Application Continued
A tourist accidentally drops her lip balm off the Golden Gate Bridge. The bridge is 220 feet from the water of the bay. The height of the lip balm is given by the polynomial –16t , where t is time in seconds. How far above the water will the lip balm be after 3 seconds?
After 3 seconds the lip balm will be 76 feet from the water.

136. Check It Out! Example 5
What if…? Another firework with a 5-second fuse is launched from the same platform at a speed of 400 feet per second. Its height is given by –16t2 +400t + 6. How high will this firework be when it explodes?
Substitute the time t to find the firework’s height.
–16t t + 6
–16(5) (5) + 6
The time is 5 seconds.
–16(25) + 400(5) + 6
Evaluate the polynomial by using the order of operations. – –
1606

137. Check It Out! Example 5 Continued
What if…? Another firework with a 5-second fuse is launched from the same platform at a speed of 400 feet per second. Its height is given by –16t2 +400t + 6. How high will this firework be when it explodes?
When the firework explodes, it will be 1606 feet above the ground.

138. Lesson Quiz: Part I
Find the degree of each polynomial.
1. 7a3b2 – 2a4 + 4b – 15
2. 25x2 – 3x4
Write each polynomial in standard form. Then give the leading coefficient.
3. 24g g5 – g2
4. 14 – x4 + 3x2 5 4
7g5 + 24g3 – g2 + 10; 7
–x4 + 3x2 + 14; –1

139. Lesson Quiz: Part II
Classify each polynomial according to its degree and number of terms.
5. 18x2 – 12x + 5
quadratic trinomial
6. 2x4 – 1
quartic binomial
7. The polynomial 3.675v v2 is used to estimate the stopping distance in feet for a car whose speed is y miles per hour on flat dry pavement. What is the stopping distance for a car traveling at 70 miles per hour? ft

140. Warm Up Simplify each expression by combining like terms. 1. 4x + 2x
2. 3y + 7y
3. 8p – 5p
4. 5n + 6n2
Simplify each expression.
5. 3(x + 4)
6. –2(t + 3)
7. –1(x2 – 4x – 6) 6x
10y 3p
not like terms
3x + 12
–2t – 6
–x2 + 4x + 6

141. Objective
Add and subtract polynomials.

142. Just as you can perform operations on numbers, you can perform operations on polynomials. To add or subtract polynomials, combine like terms.

143. Example 1: Adding and Subtracting Monomials
Add or Subtract..
A. 12p3 + 11p2 + 8p3
12p3 + 11p2 + 8p3
Identify like terms.
Rearrange terms so that like terms are together.
12p3 + 8p3 + 11p2
20p3 + 11p2
Combine like terms.
B. 5x2 – 6 – 3x + 8
Identify like terms.
5x2 – 6 – 3x + 8
Rearrange terms so that like terms are together.
5x2 – 3x + 8 – 6
5x2 – 3x + 2
Combine like terms.

144. Example 1: Adding and Subtracting Monomials
Add or Subtract..
C. t2 + 2s2 – 4t2 – s2
t2 + 2s2 – 4t2 – s2
Identify like terms.
Rearrange terms so that like terms are together.
t2 – 4t2 + 2s2 – s2
–3t2 + s2
Combine like terms.
D. 10m2n + 4m2n – 8m2n
10m2n + 4m2n – 8m2n
Identify like terms.
6m2n
Combine like terms.

145. Like terms are constants or terms with the same variable(s) raised to the same power(s). To review combining like terms, see lesson 1-7.
Remember!

146. Check It Out! Example 1
Add or subtract.
a. 2s2 + 3s2 + s
2s2 + 3s2 + s
Identify like terms.
5s2 + s
Combine like terms.
b. 4z4 – z4 + 2
4z4 – z4 + 2
Identify like terms.
Rearrange terms so that like terms are together.
4z4 + 16z4 – 8 + 2
20z4 – 6
Combine like terms.

147. Check It Out! Example 1
Add or subtract.
c. 2x8 + 7y8 – x8 – y8
Identify like terms.
2x8 + 7y8 – x8 – y8
Rearrange terms so that like terms are together.
2x8 – x8 + 7y8 – y8
x8 + 6y8
Combine like terms.
d. 9b3c2 + 5b3c2 – 13b3c2
9b3c2 + 5b3c2 – 13b3c2
Identify like terms.
b3c2
Combine like terms.

148. Polynomials can be added in either vertical or horizontal form.
In vertical form, align the like terms and add:
In horizontal form, use the Associative and Commutative Properties to regroup and combine like terms.
5x2 + 4x + 1
+ 2x2 + 5x + 2
7x2 + 9x + 3
(5x2 + 4x + 1) + (2x2 + 5x + 2)
= (5x2 + 2x2 + 1) + (4x + 5x) + (1 + 2)
= 7x2 + 9x + 3

149. Example 2: Adding Polynomials
A. (4m2 + 5) + (m2 – m + 6)
(4m2 + 5) + (m2 – m + 6)
Identify like terms.
Group like terms together.
(4m2 + m2) + (–m) +(5 + 6)
5m2 – m + 11
Combine like terms.
B. (10xy + x) + (–3xy + y)
(10xy + x) + (–3xy + y)
Identify like terms.
Group like terms together.
(10xy – 3xy) + x + y
7xy + x + y
Combine like terms.

150. Example 2C: Adding Polynomials
(6x2 – 4y) + (3x2 + 3y – 8x2 – 2y)
(6x2 – 4y) + (3x2 + 3y – 8x2 – 2y)
Identify like terms.
Group like terms together within each polynomial.
(6x2 + 3x2 – 8x2) + (3y – 4y – 2y)
6x2 – 4y
+ –5x2 + y
Use the vertical method.
Combine like terms.
x2 – 3y
Simplify.

151. Example 2D: Adding Polynomials
Identify like terms.
Group like terms together.
Combine like terms.

152. Check It Out! Example 2
Add (5a3 + 3a2 – 6a + 12a2) + (7a3 – 10a).
(5a3 + 3a2 – 6a + 12a2) + (7a3 – 10a)
Identify like terms.
(5a3 + 7a3) + (3a2 + 12a2) + (–10a – 6a)
Group like terms together.
12a3 + 15a2 – 16a
Combine like terms.

153. To subtract polynomials, remember that subtracting is the same as adding the opposite. To find the opposite of a polynomial, you must write the opposite of each term in the polynomial:
–(2x3 – 3x + 7)= –2x3 + 3x – 7

154. Example 3A: Subtracting Polynomials
(x3 + 4y) – (2x3)
Rewrite subtraction as addition of the opposite.
(x3 + 4y) + (–2x3)
(x3 + 4y) + (–2x3)
Identify like terms.
(x3 – 2x3) + 4y
Group like terms together.
–x3 + 4y
Combine like terms.

155. Example 3B: Subtracting Polynomials
(7m4 – 2m2) – (5m4 – 5m2 + 8)
(7m4 – 2m2) + (–5m4 + 5m2 – 8)
Rewrite subtraction as addition of the opposite.
(7m4 – 2m2) + (–5m4 + 5m2 – 8)
Identify like terms.
Group like terms together.
(7m4 – 5m4) + (–2m2 + 5m2) – 8
2m4 + 3m2 – 8
Combine like terms.

156. Example 3C: Subtracting Polynomials
(–10x2 – 3x + 7) – (x2 – 9)
(–10x2 – 3x + 7) + (–x2 + 9)
Rewrite subtraction as addition of the opposite.
(–10x2 – 3x + 7) + (–x2 + 9)
Identify like terms.
–10x2 – 3x + 7
–x2 + 0x + 9
Use the vertical method.
Write 0x as a placeholder.
–11x2 – 3x + 16
Combine like terms.

157. Example 3D: Subtracting Polynomials
(9q2 – 3q) – (q2 – 5)
Rewrite subtraction as addition of the opposite.
(9q2 – 3q) + (–q2 + 5)
(9q2 – 3q) + (–q2 + 5)
Identify like terms.
Use the vertical method.
9q2 – 3q + 0
+ − q2 – 0q + 5
Write 0 and 0q as placeholders.
8q2 – 3q + 5
Combine like terms.

158. Check It Out! Example 3
Subtract.
(2x2 – 3x2 + 1) – (x2 + x + 1)
Rewrite subtraction as addition of the opposite.
(2x2 – 3x2 + 1) + (–x2 – x – 1)
(2x2 – 3x2 + 1) + (–x2 – x – 1)
Identify like terms.
Use the vertical method.
–x2 + 0x + 1
+ –x2 – x – 1
Write 0x as a placeholder.
–2x2 – x
Combine like terms.

159. Example 4: Application
A farmer must add the areas of two plots of land to determine the amount of seed to plant. The area of plot A can be represented by 3x2 + 7x – 5 and the area of plot B can be represented by 5x2 – 4x Write a polynomial that represents the total area of both plots of land.
(3x2 + 7x – 5)
Plot A. +
(5x2 – 4x + 11)
Plot B.
8x2 + 3x + 6
Combine like terms.

160. Check It Out! Example 4
The profits of two different manufacturing plants can be modeled as shown, where x is the number of units produced at each plant.
Use the information above to write a polynomial that represents the total profits from both plants.
–0.03x2 + 25x – 1500
Eastern plant profit. +
–0.02x2 + 21x – 1700
Southern plant profit.
–0.05x2 + 46x – 3200
Combine like terms.

161. Lesson Quiz: Part I
Add or subtract.
1. 7m2 + 3m + 4m2
2. (r2 + s2) – (5r2 + 4s2)
3. (10pq + 3p) + (2pq – 5p + 6pq)
4. (14d2 – 8) + (6d2 – 2d +1)
11m2 + 3m
(–4r2 – 3s2)
18pq – 2p
20d2 – 2d – 7
5. (2.5ab + 14b) – (–1.5ab + 4b)
4ab + 10b

162. Lesson Quiz: Part II
6. A painter must add the areas of two walls to determine the amount of paint needed. The area of the first wall is modeled by 4x2 + 12x + 9, and the area of the second wall is modeled by
36x2 – 12x + 1. Write a polynomial that represents the total area of the two walls.
40x2 + 10

163. Warm Up
Evaluate.
1. 32
3. 102
Simplify.
4. 23  24
6. (53)2 9
2. 24 16
100 27
5. y5  y4 y9 56
7. (x2)4 x8
8. –4(x – 7)
–4x + 28

164. Objective
Multiply polynomials.

165. To multiply monomials and polynomials, you will use some of the properties of exponents that you learned earlier in this chapter.

166. Example 1: Multiplying Monomials
A. (6y3)(3y5)
(6y3)(3y5)
Group factors with like bases together.
(6 3)(y3 y5) 
18y8
Multiply.
B. (3mn2) (9m2n)
(3mn2)(9m2n)
Group factors with like bases together.
(3 9)(m m2)(n2  n) 
27m3n3
Multiply.

167. Example 1C: Multiplying Monomials( ) æ ç è - 2 1 12 4 t s ö ÷ ø
Group factors with like bases together. ( ) g æ - ö ç è 2 1 12 4 t s ÷ ø
Multiply.

168. When multiplying powers with the same base, keep the base and add the exponents.
x2  x3 = x2+3 = x5
Remember!

169. Group factors with like bases together. (3x3)(6x2)
Check It Out! Example 1
Multiply.
a. (3x3)(6x2)
Group factors with like bases together.
(3x3)(6x2)
(3 6)(x3 x2) 
Multiply.
18x5
b. (2r2t)(5t3)
Group factors with like bases together.
(2r2t)(5t3)
(2 5)(r2)(t3 t) 
Multiply.
10r2t4

170. ( ) ( ) ( ) ( ) Check It Out! Example 1 Multiply. æ 1 ö c. x y 12 x z3 2 4 5 ç ÷
y z è 3 ø ( ) æ ç è 4 5 2 1 12 3 x z y ö ÷ ø
Group factors with like bases together. ( ) g æ ç è 3 2 4 5 1 12 z
x x
y y ö ÷ ø
Multiply. 7 5 4 x y z

171. To multiply a polynomial by a monomial, use the Distributive Property.

172. Example 2A: Multiplying a Polynomial by a Monomial
4(3x2 + 4x – 8)
4(3x2 + 4x – 8)
Distribute 4.
(4)3x2 +(4)4x – (4)8
Multiply.
12x2 + 16x – 32

173. Example 2B: Multiplying a Polynomial by a Monomial
6pq(2p – q)
(6pq)(2p – q)
Distribute 6pq.
(6pq)2p + (6pq)(–q)
Group like bases together.
(6  2)(p  p)(q) + (–1)(6)(p)(q  q)
12p2q – 6pq2
Multiply.

174. Example 2C: Multiplying a Polynomial by a Monomial1 ( )
x y 2 6 xy + 8
x y 2 2 2
x y ( ) + 2 6 1 xy y x 8
Distribute 2 1
x y
x y
x y ( ) æ ç è + 2 1 6 8 xy ö ÷ ø
Group like bases together.
x2 • x ( ) æ + ç è 1
• 6 2
y • y
x2 • x2
y • y2
• 8 ö ÷ ø
3x3y2 + 4x4y3
Multiply.

175. Check It Out! Example 2
Multiply.
a. 2(4x2 + x + 3)
2(4x2 + x + 3)
Distribute 2.
2(4x2) + 2(x) + 2(3)
Multiply.
8x2 + 2x + 6

176. Check It Out! Example 2
Multiply.
b. 3ab(5a2 + b)
3ab(5a2 + b)
Distribute 3ab.
(3ab)(5a2) + (3ab)(b)
Group like bases together.
(3  5)(a  a2)(b) + (3)(a)(b  b)
15a3b + 3ab2
Multiply.

177. Check It Out! Example 2
Multiply.
c. 5r2s2(r – 3s)
5r2s2(r – 3s)
Distribute 5r2s2.
(5r2s2)(r) – (5r2s2)(3s)
Group like bases together.
(5)(r2  r)(s2) – (5  3)(r2)(s2  s)
5r3s2 – 15r2s3
Multiply.

178. To multiply a binomial by a binomial, you can apply the Distributive Property more than once:
(x + 3)(x + 2) = x(x + 2) + 3(x + 2)
Distribute x and 3.
Distribute x and 3 again.
= x(x + 2) + 3(x + 2)
= x(x) + x(2) + 3(x) + 3(2)
Multiply.
= x2 + 2x + 3x + 6
Combine like terms.
= x2 + 5x + 6

179. Another method for multiplying binomials is called the FOIL method.
1. Multiply the First terms. (x + 3)(x + 2) x x = x2  O
2. Multiply the Outer terms. (x + 3)(x + 2) x 2 = 2x  I
3. Multiply the Inner terms. (x + 3)(x + 2) x = 3x  L
4. Multiply the Last terms. (x + 3)(x + 2) = 6 
(x + 3)(x + 2) = x2 + 2x + 3x + 6 = x2 + 5x + 6 F O I L

180. Example 3A: Multiplying Binomials
(s + 4)(s – 2)
(s + 4)(s – 2)
s(s – 2) + 4(s – 2)
Distribute s and 4.
s(s) + s(–2) + 4(s) + 4(–2)
Distribute s and 4 again.
s2 – 2s + 4s – 8
Multiply.
s2 + 2s – 8
Combine like terms.

181. Example 3B: Multiplying Binomials
Write as a product of two binomials.
(x – 4)2
(x – 4)(x – 4)
Use the FOIL method.
(x x) + (x (–4)) + (–4  x) + (–4  (–4)) 
x2 – 4x – 4x + 8
Multiply.
x2 – 8x + 8
Combine like terms.

182. Example 3C: Multiplying Binomials
(8m2 – n)(m2 – 3n)
Use the FOIL method.
8m2(m2) + 8m2(–3n) – n(m2) – n(–3n)
8m4 – 24m2n – m2n + 3n2
Multiply.
8m4 – 25m2n + 3n2
Combine like terms.

183. In the expression (x + 5)2, the base is (x + 5)
In the expression (x + 5)2, the base is (x + 5). (x + 5)2 = (x + 5)(x + 5)
Helpful Hint

184. Check It Out! Example 3a
Multiply.
(a + 3)(a – 4)
(a + 3)(a – 4)
Distribute a and 3.
a(a – 4)+3(a – 4)
Distribute a and 3 again.
a(a) + a(–4) + 3(a) + 3(–4)
a2 – 4a + 3a – 12
Multiply.
a2 – a – 12
Combine like terms.

185. Write as a product of two binomials. (x – 3)2
Check It Out! Example 3b
Multiply.
Write as a product of two binomials.
(x – 3)2
(x – 3)(x – 3)
Use the FOIL method.
(x x) + (x(–3)) + (–3  x)+ (–3)(–3) ●
x2 – 3x – 3x + 9
Multiply.
x2 – 6x + 9
Combine like terms.

186. Check It Out! Example 3c
Multiply.
(2a – b2)(a + 4b2)
(2a – b2)(a + 4b2)
Use the FOIL method.
2a(a) + 2a(4b2) – b2(a) + (–b2)(4b2)
2a2 + 8ab2 – ab2 – 4b4
Multiply.
2a2 + 7ab2 – 4b4
Combine like terms.

187. To multiply polynomials with more than two terms, you can use the Distributive Property several times. Multiply (5x + 3) by (2x2 + 10x – 6):
(5x + 3)(2x2 + 10x – 6) = 5x(2x2 + 10x – 6) + 3(2x2 + 10x – 6)
= 5x(2x2 + 10x – 6) + 3(2x2 + 10x – 6)
= 5x(2x2) + 5x(10x) + 5x(–6) + 3(2x2) + 3(10x) + 3(–6)
= 10x3 + 50x2 – 30x + 6x2 + 30x – 18
= 10x3 + 56x2 – 18

188. You can also use a rectangle model to multiply polynomials with more than two terms. This is similar to finding the area of a rectangle with length (2x2 + 10x – 6) and width (5x + 3):
2x2
+10x –6
10x3
50x2
–30x
30x
6x2
–18 5x +3
Write the product of the monomials in each row and column:
To find the product, add all of the terms inside the rectangle by combining like terms and simplifying if necessary.
10x3 + 6x2 + 50x2 + 30x – 30x – 18
10x3 + 56x2 – 18

189. Another method that can be used to multiply polynomials with more than two terms is the vertical method. This is similar to methods used to multiply whole numbers.
Multiply each term in the top polynomial by 3.
2x2 + 10x – 6
5x + 3 
Multiply each term in the top polynomial by 5x, and align like terms.
6x2 + 30x – 18
+ 10x3 + 50x2 – 30x
Combine like terms by adding vertically.
10x3 + 56x x – 18
10x3 + 56x – 18
Simplify.

190. Example 4A: Multiplying Polynomials
(x – 5)(x2 + 4x – 6)
(x – 5 )(x2 + 4x – 6)
Distribute x and –5.
x(x2 + 4x – 6) – 5(x2 + 4x – 6)
Distribute x and −5 again.
x(x2) + x(4x) + x(–6) – 5(x2) – 5(4x) – 5(–6)
x3 + 4x2 – 5x2 – 6x – 20x + 30
Simplify.
x3 – x2 – 26x + 30
Combine like terms.

191. Example 4B: Multiplying Polynomials
(2x – 5)(–4x2 – 10x + 3)
Multiply each term in the top polynomial by –5.
(2x – 5)(–4x2 – 10x + 3)
Multiply each term in the top polynomial by 2x, and align like terms.
–4x2 – 10x + 3
2x – 5 x
20x2 + 50x – 15
+ –8x3 – 20x2 + 6x
Combine like terms by adding vertically.
–8x x – 15

192. Example 4C: Multiplying Polynomials
[(x + 3)(x + 3)](x + 3)
Write as the product of three binomials.
[x(x+3) + 3(x+3)](x + 3)
Use the FOIL method on the first two factors.
(x2 + 3x + 3x + 9)(x + 3)
Multiply.
(x2 + 6x + 9)(x + 3)
Combine like terms.

193. Example 4C: Multiplying Polynomials
Use the Commutative Property of Multiplication.
(x + 3)(x2 + 6x + 9)
x(x2 + 6x + 9) + 3(x2 + 6x + 9)
Distribute the x and 3.
x(x2) + x(6x) + x(9) + 3(x2) + 3(6x) + 3(9)
Distribute the x and 3 again.
x3 + 6x2 + 9x + 3x2 + 18x + 27
Combine like terms.
x3 + 9x2 + 27x + 27

194. Example 4D: Multiplying Polynomials
(3x + 1)(x3 – 4x2 – 7)
Write the product of the monomials in each row and column. x3
4x2 –7 3x
3x4
12x3
–21x +1
4x2 x3 –7
Add all terms inside the rectangle.
3x4 + 12x3 + x3 + 4x2 – 21x – 7
3x4 + 13x3 + 4x2 – 21x – 7
Combine like terms.

195. A polynomial with m terms multiplied by a polynomial with n terms has a product that, before simplifying has mn terms. In Example 4A, there are 2 3, or 6 terms before simplifying.
Helpful Hint 

196. Check It Out! Example 4a
Multiply.
(x + 3)(x2 – 4x + 6)
(x + 3 )(x2 – 4x + 6)
Distribute x and 3.
x(x2 – 4x + 6) + 3(x2 – 4x + 6)
Distribute x and 3 again.
x(x2) + x(–4x) + x(6) +3(x2) +3(–4x) +3(6)
x3 – 4x2 + 3x2 +6x – 12x + 18
Simplify.
x3 – x2 – 6x + 18
Combine like terms.

197. Check It Out! Example 4b
Multiply.
(3x + 2)(x2 – 2x + 5)
Multiply each term in the top polynomial by 2.
(3x + 2)(x2 – 2x + 5)
Multiply each term in the top polynomial by 3x, and align like terms.
x2 – 2x + 5
3x + 2 
2x2 – 4x + 10
+ 3x3 – 6x2 + 15x
3x3 – 4x2 + 11x + 10
Combine like terms by adding vertically.

198. Write the formula for the area of a rectangle.
Example 5: Application
The width of a rectangular prism is 3 feet less than the height, and the length of the prism is 4 feet more than the height.
a. Write a polynomial that represents the area of the base of the prism.
A = l  w
A = l w 
Write the formula for the area of a rectangle.
Substitute h – 3 for w and h + 4 for l.
A = (h + 4)(h – 3)
A = h2 + 4h – 3h – 12
Multiply.
A = h2 + h – 12
Combine like terms.
The area is represented by h2 + h – 12.

199. Example 5: Application
The width of a rectangular prism is 3 feet less than the height, and the length of the prism is 4 feet more than the height.
b. Find the area of the base when the height is 5 ft.
A = h2 + h – 12
Write the formula for the area the base of the prism.
A = h2 + h – 12
A = – 12
Substitute 5 for h.
A = – 12
Simplify.
A = 18
Combine terms.
The area is 18 square feet.

200. The length of a rectangle is 4 meters shorter than its width.
Check It Out! Example 5
The length of a rectangle is 4 meters shorter than its width.
a. Write a polynomial that represents the area of the rectangle.
A = l w 
A = l w
Write the formula for the area of a rectangle.
Substitute x – 4 for l and x for w.
A = x(x – 4)
A = x2 – 4x
Multiply.
The area is represented by x2 – 4x.

201. Check It Out! Example 5
The length of a rectangle is 4 meters shorter than its width.
b. Find the area of a rectangle when the width is 6 meters.
A = x2 – 4x
Write the formula for the area of a rectangle whose length is 4 meters shorter than width .
A = x2 – 4x
A = 62 – 4  6
Substitute 6 for x.
A = 36 – 24
Simplify.
A = 12
Combine terms.
The area is 12 square meters.

202. Lesson Quiz: Part I
Multiply.
1. (6s2t2)(3st)
2. 4xy2(x + y)
3. (x + 2)(x – 8)
4. (2x – 7)(x2 + 3x – 4)
5. 6mn(m2 + 10mn – 2)
6. (2x – 5y)(3x + y)
18s3t3
4x2y2 + 4xy3
x2 – 6x – 16
2x3 – x2 – 29x + 28
6m3n + 60m2n2 – 12mn
6x2 – 13xy – 5y2

203. Lesson Quiz: Part II
7. A triangle has a base that is 4cm longer than its height.
a. Write a polynomial that represents the area of the triangle. 1 2
h2 + 2h
b. Find the area when the height is 8 cm.
48 cm2

204. Warm Up
Simplify.
1. 42
3. (–2) (x)2
5. –(5y2) 16
2. 72 49 4 x2
–25y2
6. (m2)2 m4
7. 2(6xy)
12xy 8.
2(8x2)
16x2

205. Objective
Find special products of binomials.

206. Vocabulary
perfect-square trinomial
difference of two squares

207. Imagine a square with sides of length (a + b):
The area of this square is (a + b)(a + b) or (a + b)2. The area of this square can also be found by adding the areas of the smaller squares and the rectangles inside. The sum of the areas inside is a2 + ab + ab + b2.

208. This means that (a + b)2 = a2+ 2ab + b2.
You can use the FOIL method to verify this: F L
(a + b)2 = (a + b)(a + b) = a2 + ab + ab + b2 I
= a2 + 2ab + b2 O
A trinomial of the form a2 + 2ab + b2 is called a perfect-square trinomial. A perfect-square trinomial is a trinomial that is the result of squaring a binomial.

209. Example 1: Finding Products in the Form (a + b)2
Multiply.
A. (x +3)2
Use the rule for (a + b)2.
(a + b)2 = a2 + 2ab + b2
Identify a and b: a = x and b = 3.
(x + 3)2 = x2 + 2(x)(3) + 32
= x2 + 6x + 9
Simplify.
B. (4s + 3t)2
Use the rule for (a + b)2.
(a + b)2 = a2 + 2ab + b2
Identify a and b: a = 4s and b = 3t.
(4s + 3t)2 = (4s)2 + 2(4s)(3t) + (3t)2
= 16s2 + 24st + 9t2
Simplify.

210. Example 1C: Finding Products in the Form (a + b)2
Multiply.
C. (5 + m2)2
Use the rule for (a + b)2.
(a + b)2 = a2 + 2ab + b2
Identify a and b: a = 5 and b = m2.
(5 + m2)2 = (5)(m2) + (m2)2
= m2 + m4
Simplify.

211. Check It Out! Example 1
Multiply.
A. (x + 6)2
Use the rule for (a + b)2.
(a + b)2 = a2 + 2ab + b2
Identify a and b: a = x and b = 6.
(x + 6)2 = x2 + 2(x)(6) + 62
= x2 + 12x + 36
Simplify.
B. (5a + b)2
Use the rule for (a + b)2.
(a + b)2 = a2 + 2ab + b2
Identify a and b: a = 5a and b = b.
(5a + b)2 = (5a)2 + 2(5a)(b) + b2
= 25a2 + 10ab + b2
Simplify.

212. Check It Out! Example 1C
Multiply.
Use the rule for (a + b)2.
(1 + c3)2
(a + b)2 = a2 + 2ab + b2
Identify a and b: a = 1 and b = c3.
(1 + c3)2 = (1)(c3) + (c3)2
Simplify.
= 1 + 2c3 + c6

213. You can use the FOIL method to find products in the form of (a – b)2.
(a – b)2 = (a – b)(a – b) = a2 – ab – ab + b2 I O
= a2 – 2ab + b2
A trinomial of the form a2 – ab + b2 is also a perfect-square trinomial because it is the result of squaring the binomial (a – b).

214. Example 2: Finding Products in the Form (a – b)2
Multiply.
A. (x – 6)2
Use the rule for (a – b)2.
(a – b)2 = a2 – 2ab + b2
Identify a and b: a = x and b = 6.
(x – 6)2 = x2 – 2x(6) + (6)2
= x2 – 12x + 36
Simplify.
B. (4m – 10)2
Use the rule for (a – b)2.
Identify a and b: a = 4m and b = 10.
(a – b)2 = a2 – 2ab + b2
(4m – 10)2 = (4m)2 – 2(4m)(10) + (10)2
= 16m2 – 80m + 100
Simplify.

215. Example 2: Finding Products in the Form (a – b)2
Multiply.
C. (2x – 5y)2
Use the rule for (a – b)2.
(a – b)2 = a2 – 2ab + b2
Identify a and b: a = 2x and b = 5y.
(2x – 5y)2 = (2x)2 – 2(2x)(5y) + (5y)2
= 4x2 – 20xy +25y2
Simplify.
D. (7 – r3)2
Use the rule for (a – b)2.
(a – b)2 = a2 – 2ab + b2
Identify a and b: a = 7 and b = r3.
(7 – r3)2 = 72 – 2(7)(r3) + (r3)2
= 49 – 14r3 + r6
Simplify.

216. Check It Out! Example 2
Multiply.
a. (x – 7)2
Use the rule for (a – b)2.
(a – b)2 = a2 – 2ab + b2
Identify a and b: a = x and b = 7.
(x – 7)2 = x2 – 2(x)(7) + (7)2
= x2 – 14x + 49
Simplify.
b. (3b – 2c)2
Use the rule for (a – b)2.
(a – b)2 = a2 – 2ab + b2
Identify a and b: a = 3b and b = 2c.
(3b – 2c)2 = (3b)2 – 2(3b)(2c) + (2c)2
= 9b2 – 12bc + 4c2
Simplify.

217. Check It Out! Example 2c
Multiply.
(a2 – 4)2
Use the rule for (a – b)2.
(a – b)2 = a2 – 2ab + b2
Identify a and b: a = a2 and b = 4.
(a2 – 4)2 = (a2)2 – 2(a2)(4) + (4)2
= a4 – 8a2 + 16
Simplify.

218. You can use an area model to see that
(a + b)(a–b)= a2 – b2.
Begin with a square with area a2. Remove a square with area b2. The area of the new figure is a2 – b2.
Remove the rectangle on the bottom. Turn it and slide it up next to the top rectangle.
The new arrange- ment is a rectangle with length a + b and width a – b. Its area is (a + b)(a – b).
So (a + b)(a – b) = a2 – b2. A binomial of the form a2 – b2 is called a difference of two squares.

219. Example 3: Finding Products in the Form (a + b)(a – b)
Multiply.
A. (x + 4)(x – 4)
Use the rule for (a + b)(a – b).
(a + b)(a – b) = a2 – b2
(x + 4)(x – 4) = x2 – 42
Identify a and b: a = x and b = 4.
= x2 – 16
Simplify.
B. (p2 + 8q)(p2 – 8q)
(a + b)(a – b) = a2 – b2
Use the rule for (a + b)(a – b).
(p2 + 8q)(p2 – 8q) = (p2)2 – (8q)2
Identify a and b: a = p2 and b = 8q.
= p4 – 64q2
Simplify.

220. Example 3: Finding Products in the Form (a + b)(a – b)
Multiply.
C. (10 + b)(10 – b)
Use the rule for (a + b)(a – b).
(a + b)(a – b) = a2 – b2
Identify a and b: a = 10 and b = b.
(10 + b)(10 – b) = 102 – b2
= 100 – b2
Simplify.

221. Check It Out! Example 3
Multiply.
a. (x + 8)(x – 8)
Use the rule for (a + b)(a – b).
(a + b)(a – b) = a2 – b2
Identify a and b: a = x and b = 8.
(x + 8)(x – 8) = x2 – 82
= x2 – 64
Simplify.
b. (3 + 2y2)(3 – 2y2)
Use the rule for (a + b)(a – b).
(a + b)(a – b) = a2 – b2
Identify a and b: a = 3 and b = 2y2.
(3 + 2y2)(3 – 2y2) = 32 – (2y2)2
= 9 – 4y4
Simplify.

222. Check It Out! Example 3
Multiply.
c. (9 + r)(9 – r)
Use the rule for (a + b)(a – b).
(a + b)(a – b) = a2 – b2
Identify a and b: a = 9 and b = r.
(9 + r)(9 – r) = 92 – r2
= 81 – r2
Simplify.

223. Example 4: Problem-Solving Application
Write a polynomial that represents the area of the yard around the pool shown below.

224. Understand the Problem
Example 4 Continued 1
Understand the Problem
The answer will be an expression that shows the area of the yard less the area of the pool.
List important information:
The yard is a square with a side length of x + 5.
The pool has side lengths of x + 2 and x – 2.

225. Example 4 Continued 2
Make a Plan
The area of the yard is (x + 5)2. The area of the pool is (x + 2) (x – 2). You can subtract the area of the pool from the yard to find the area of the yard surrounding the pool.

226. Example 4 Continued
Solve 3
Step 1 Find the total area.
(x +5)2 = x2 + 2(x)(5) + 52
Use the rule for (a + b)2: a = x and b = 5.
x2 + 10x + 25 =
Step 2 Find the area of the pool.
(x + 2)(x – 2) = x2 – 2x + 2x – 4
Use the rule for (a + b)(a – b): a = x and b = 2.
x2 – 4 =

227. Example 4 Continued
Solve 3
Step 3
Find the area of the yard around the pool.
Area of yard =
total area –
area of pool a =
x2 + 10x + 25
(x2 – 4) –
Identify like terms.
= x2 + 10x + 25 – x2 + 4
= (x2 – x2) + 10x + ( )
Group like terms together
= 10x + 29
The area of the yard around the pool is 10x + 29.

228.  Example 4 Continued 4 Look Back
Suppose that x = 20. Then the total area in the back yard would be 252 or 625. The area of the pool would be 22  18 or 396. The area of the yard around the pool would be 625 – 396 = 229.
According to the solution, the area of the yard around the pool is 10x If x = 20, then 10x +29 = 10(20) + 29 = 229. 

229. To subtract a polynomial, add the opposite of each term.
Remember!

230. Check It Out! Example 4
Write an expression that represents the area of the swimming pool.

231. Understand the Problem
Check It Out! Example 4 Continued 1
Understand the Problem
The answer will be an expression that shows the area of the two rectangles combined.
List important information:
The upper rectangle has side lengths of 5 + x and 5 – x .
The lower rectangle is a square with side length of x.

232. Check It Out! Example 4 Continued2
Make a Plan
The area of the upper rectangle is (5 + x)(5 – x). The area of the lower square is x2. Added together they give the total area of the pool.

233. Check It Out! Example 4 Continued
Solve 3
Step 1 Find the area of the upper rectangle.
Use the rule for (a + b)
(a – b): a = 5 and b = x.
(5 + x)(5 – x) = 25 – 5x + 5x – x2
–x2 + 25 =
Step 2 Find the area of the lower square.
= x x  x2 =

234. Check It Out! Example 4 Continued
Solve 3
Step 3
Find the area of the pool.
Area of pool =
rectangle area +
square area a =
–x2 + 25 + x2
= –x x2
Identify like terms.
= (x2 – x2) + 25
Group like terms together
= 25
The area of the pool is 25.

235. Check It Out! Example 4 Continued
Look Back
Suppose that x = 2. Then the area of the upper rectangle would be 21. The area of the lower square would be 4. The area of the pool would be = 25.
According to the solution, the area of the pool is 25. 

237. Lesson Quiz: Part I
Multiply.
1. (x + 7)2
2. (x – 2)2
3. (5x + 2y)2
4. (2x – 9y)2
5. (4x + 5y)(4x – 5y)
6. (m2 + 2n)(m2 – 2n)
x2 + 14x + 49
x2 – 4x + 4
25x2 + 20xy + 4y2
4x2 – 36xy + 81y2
16x2 – 25y2
m4 – 4n2

238. Lesson Quiz: Part II
7. Write a polynomial that represents the shaded area of the figure below.
x + 6
x – 7
x – 6
x – 7
14x – 85