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Exploring Polynomials
LESSON 12-1
Course 3
Problem of the Day
The telephone company is installing telephone lines for ten buildings. Each
building is to be connected to each of the other buildings with one line. How
many telephone lines are needed?
45
Lesson
Main
Lesson
12-1
Feature
Exploring Polynomials
LESSON 12-1
Course 3
Check Skills You’ll Need
(For help, go to Lesson 6-2.)
1.
Vocabulary Review Are x,
1
x, and – 4x like terms?
2
Simplify each expression.
2.
–2 + 2t – 3t
3.
7w – 10 + 5w
4.
3k + 32k – 5
Check Skills You’ll Need
Lesson
Main
Lesson
12-1
Feature
Exploring Polynomials
LESSON 12-1
Course 3
Check Skills You’ll Need
Solutions
1. yes; all the terms include the same variable, x
2.
–2 + 2t – 3t = –2 + (2t – 3t)
= –2 + (2 – 3)t
= –2 – t
3. 7w – 10 + 5w = 7w + 5w – 10
= (7w + 5w) – 10
= (7 + 5)w – 10
= 12w – 10
4. 3k + 32k – 5 = (3k + 32k) – 5
= (3 + 32)k – 5
= 35k – 5
Lesson
Main
Lesson
12-1
Feature
Exploring Polynomials
LESSON 12-1
Course 3
Additional Examples
Write a variable expression for this model.
The model shows the expression x2 – 2x + 3.
Quick Check
Lesson
Main
Lesson
12-1
Feature
Exploring Polynomials
LESSON 12-1
Course 3
Additional Examples
Use tiles to simplify the polynomial
–x2 + 3x + x2 + x2 + 3 – x – 4.
Model each term.
Group like terms together. Remove zero pairs.
The simplified polynomial is x2 + 2x – 1.
Lesson
Main
Lesson
12-1
Quick Check
Feature
Exploring Polynomials
LESSON 12-1
Course 3
Additional Examples
A playground has areas of grass and sand. The polynomial
4x2 – 5 + 2x + 2x2 + 3 – 5x represents the total area of the grass minus
the sandy areas. Use properties of numbers to simplify the polynomial.
4x2 – 5 + 2x + 2x2 + 3 – 5x
= 4x2 + 2x2 +2x – 5x + 3 – 5
Use the Commutative Property.
= (4x2 + 2x2 ) +(2x – 5x) + (3 – 5)
Use the Associative Property.
= (4 + 2)x2 +(2 – 5)x + (3 – 5)
Use the Distributive Property.
= 6x2 – 3x – 2
The total area of the grass can be represented by 6x2 – 3x – 2.
Quick Check
Lesson
Main
Lesson
12-1
Feature
Exploring Polynomials
LESSON 12-1
Course 3
Lesson Quiz
1. Write a variable expression for the model.
x2 – 2x + 4
2. Write a variable expression for the model.
– x2 + 3x – 1
3. Use properties to simplify the polynomial 4x2 + 9x – x2 – 10x – 2.
3x2 – x – 2
4. Use properties to simplify the polynomial 5a2 – 7a + 4 + a – 7a2 + 6.
– 2a2 – 6a + 10
Lesson
Main
Lesson
12-1
Feature
Adding and Subtracting Polynomials
LESSON 12-2
Course 3
Problem of the Day
Bianca’s family needs to choose the exterior paint for their new house. The
wall colors are white, green, and beige. The trim colors are white, green,
blue, and cocoa. How many combinations of wall color and trim are possible?
12
Lesson
Main
Lesson
12-2
Feature
Adding and Subtracting Polynomials
LESSON 12-2
Course 3
Check Skills You’ll Need
(For help, go to Lesson 12-1.)
1. Vocabulary Review Name the constant in the
polynomial 1 – p + 2p.
Simplify.
2. 2y2 + 3y + (–5y)
3. –x + x + 6x2 – 1
4. 7z – 8z2 + z + 3z2
Check Skills You’ll Need
Lesson
Main
Lesson
12-2
Feature
Adding and Subtracting Polynomials
LESSON 12-2
Course 3
Check Skills You’ll Need
Solutions
1. 1
2. 2y2 + 3y + (–5y) = 2y2 + [3 + (–5)]y = 2y2 – 2y
3. –x + x + 6x2 – 1 = (–1 + 1)x + 6x2 – 1 = 6x2 – 1
4. 7z – 8z2 + z + 3z2 = (7z + z) + (–8z2 + 3z2) = 8z – 5z2
Lesson
Main
Lesson
12-2
Feature
Adding and Subtracting Polynomials
LESSON 12-2
Course 3
Additional Examples
Add: (5p2 + 2p + 7) + (2p2 – p – 5).
Method 1 Add using tiles.
Lesson
Main
Lesson
12-2
Feature
Adding and Subtracting Polynomials
LESSON 12-2
Course 3
Additional Examples
Quick Check
(continued)
Method 2 Add using properties.
(5p2 + 2p + 7) + (2p2 – p – 5)
= (5p2 + 2p2) + (2p – p) + (7 – 5)
Group like terms.
= (5 + 2)p2 + (2 – 1)p + (7 – 5)
Use the Distributive Property.
= 7p2 + p + 2
Simplify.
Check Check the solution in Example 1 by substituting 1 for p.
(5p2 + 2p + 7) + (2p2 – p – 5)
(5 • 12 + 2 • 1 + 7) + (2 • 12 – 1 – 5)
(5 + 2 + 7) + (2 – 1 – 5)
7p2 + p + 2
(7 • 12 + 1 + 2)
Substitute 1 for p.
(7 + 1 + 2)
Multiply.
10 = 10
Lesson
Main
Lesson
12-2
Add.
Feature
Adding and Subtracting Polynomials
LESSON 12-2
Course 3
Additional Examples
Quick Check
A garden has sides of 3x + 5, 4x – 2, 5x + 2, and 7x – 6. Write
a polynomial to express the length of edging that is needed to go
around the garden.
To find the perimeter of the garden, find the sum of the four sides.
perimeter = (3x + 5) + (4x – 2) + (5x + 2) + (7x – 6)
= (3x + 4x + 5x + 7x) + (5 – 2 + 2 – 6)
Group like terms.
= 19x – 1
Add the
coefficients.
The perimeter of the garden is (19x – 1). The edging must be (19x – 1)
units long to go around the garden.
Lesson
Main
Lesson
12-2
Feature
Adding and Subtracting Polynomials
LESSON 12-2
Course 3
Additional Examples
Subtract: (3q2 – 2q + 4) – (2q2 – 2q + 3).
(3q2 – 2q + 4) – (2q2 – 2q + 3)
= (3q2 – 2q + 4) + (–2q2 + 2q – 3)
Add the opposite
of each term in the
second polynomial.
= (3q2 – 2q2) + (–2q + 2q) + (4 – 3)
Group like terms.
= (3 – 2)q2 + (–2 + 2)q + (4 – 3)
Use the Distributive Property.
= q2 + 1
Simplify.
Lesson
Main
Lesson
12-2
Quick Check
Feature
Adding and Subtracting Polynomials
LESSON 12-2
Course 3
Lesson Quiz
Simplify each polynomial.
1. (4n2 + n + 1) + (n2 + 3n + 1)
5n2 + 4n + 2
2. (x2 – 2x + 6) + (x2 + 2x – 2)
2x2 + 4
3. (a2 – 7) – (a2 + 4a – 4)
–4a – 3
4. (m – 5) + (m2 – 12) + (6m2 – 9m)
7m2 – 8m – 17
Lesson
Main
Lesson
12-2
Feature
Exponents and Multiplication
LESSON 12-3
Course 3
Problem of the Day
1
2
Find the area of a rectangle 3 ft wide and twice as high.
1
242 ft2
Lesson
Main
Lesson
12-3
Feature
Exponents and Multiplication
LESSON 12-3
Course 3
Check Skills You’ll Need
(For help, go to Lesson 2-7.)
1.
Vocabulary Review What is the base of the exponential
expression xy?
Simplify each expression.
2.
(–1)4
3.
(–3)2
4.
–32
5.
–14
Lesson
Main
Check Skills You’ll Need
Lesson
12-3
Feature
Exponents and Multiplication
LESSON 12-3
Course 3
Check Skills You’ll Need
Solutions
1. x
2. (–1)4 = (–1) • (–1) • (–1) • (–1) = 1
3. (–3)2 = (–3) • (–3) = 9
4. –32 = –1(3 • 3) = –9
5. –14 = –1(1 • 1 • 1 • 1) = –1
Lesson
Main
Lesson
12-3
Feature
Exponents and Multiplication
LESSON 12-3
Course 3
Additional Examples
Write the expression using a single exponent.
(–3)2
(–3)4
(–3)2
(–3)4 = (–3)(2+4)
= (–3)(6)
Add the exponents.
Simplify the exponent.
Quick Check
Lesson
Main
Lesson
12-3
Feature
Exponents and Multiplication
LESSON 12-3
Course 3
Additional Examples
Quick Check
Multiply (3 x 103) (7 x 105). Write the product in scientific
notation.
(3 103) (7 105) = (3 • 7) (103 • 105)
Lesson
Main
Use the Associative and
Commutative properties.
= 21 (103 • 105)
Multiply 3 and 7.
= 21
Add the exponents for
the powers of 10.
108
= 2.1 101 108
Write 21 in scientific
notation.
= 2.1 109
Add the exponents.
Lesson
12-3
Feature
Exponents and Multiplication
LESSON 12-3
Course 3
Additional Examples
A light-year is 5.9 x 1012 miles. A mile is 1.609 x 103 meters. In
scientific notation, about how many meters are in a light year?
(5.9 x 1012)
(1.609 x 103)
= (5.9 1.609) x (1012 103)
9.5 x (1012 103)
= 9.5 x 1015
Multiply by the conversion factor.
Use the Associative and Commutative
properties.
Multiply 5.9 and 1.609. Round to the
nearest tenth.
Add exponents of the powers of 10.
Quick Check
Lesson
Main
Lesson
12-3
Feature
Exponents and Multiplication
LESSON 12-3
Course 3
Lesson Quiz
1. Write (–8)4 • (–8)5 using a single exponent.
(–8)9
2. Write the product of (8.2 x 106) and (5 x 102) in scientific notation.
4.1 x 109
3. The speed of light is 3.00 x 105 km/s. Find the distance light travels in
8 x 102 seconds.
2.4 x 108 km
4. A light-year is 5.9 x 1012 miles. A mile is approximately 6.34 x 104 inches.
About how many inches are in a light-year?
3. 74 x 1017 in
Lesson
Main
Lesson
12-3
Feature
Multiplying Polynomials
LESSON 12-4
Course 3
Problem of the Day
Palindromes are numbers, words, or sentences that read the same forward
and backward. Find a number to add to each number to get a palindrome as
a sum.
175.3
60.32
0.271
1.94
Lesson
Main
Lesson
12-4
Feature
Multiplying Polynomials
LESSON 12-4
Course 3
Check Skills You’ll Need
(For help, go to Lesson 12-3.)
1. Vocabulary Review The expression 23 • 25 can be
simplified by adding the
Simplify using a single exponent.
2. x4 • x5 • x6
3. (–a)3 • (–a)7
Check Skills You’ll Need
Lesson
Main
Lesson
12-4
Feature
Multiplying Polynomials
LESSON 12-4
Course 3
Check Skills You’ll Need
Solutions
1.
exponents
2 . x4 • x5 • x6 = x (4 + 5 + 6) = x15
3 . (–a)3 • (–a)7 = (–a)(3+7) = (–a)10 = a10
Lesson
Main
Lesson
12-4
Feature
Multiplying Polynomials
LESSON 12-4
Course 3
Additional Examples
Simplify (3x2)(–4x3).
Use the Commutative Property to
rearrange the factors.
(3x2)(–4x3) = (3)(–4) • x2 • x3
= –12 • x2 • x3
Multiply the coefficients.
= –12x5
Add the exponents.
Quick Check
Lesson
Main
Lesson
12-4
Feature
Multiplying Polynomials
LESSON 12-4
Course 3
Additional Examples
If the width of a new house foundation is represented by 2x
and the length is represented by 4x + 6, which expression represents
the area of the foundation?
To find the area, multiply the width, 2x, times the length, 4x + 6.
A = l w = 2x(4x + 6)
= 2x 4x + 2x 6
= 8x2 + 12x
Quick Check
Lesson
Main
Lesson
12-4
Feature
Multiplying Polynomials
LESSON 12-4
Course 3
Additional Examples
Use an area model to simplify (x + 1)(3x + 1).
Count each type of tile.
There are three x2 tiles.
There are four x tiles.
There is one unit tile.
So, (x + 1)(3x + 1) = 3x2 + 4x + 1.
Quick Check
Lesson
Main
Lesson
12-4
Feature
Multiplying Polynomials
LESSON 12-4
Course 3
Lesson Quiz
Simplify each expression in 1–3.
1. (–7c3)(4c)
–28c4
2. x(3x + 2)
3x2 + 2x
3. –2m2(m – 1)
–2m3 + 2m2
4. If the base of a parallelogram-shaped playground is represented by 6y,
and the height is represented by 4y – 3, which expression represents
the area?
24y2 – 18y
Lesson
Main
Lesson
12-4
Feature
Exponents and Division
LESSON 12-5
Course 3
Problem of the Day
1
Lupe’s flag is a square with 1 yd on a side. Anna makes her flag
2
1 yd larger on each side. How much larger is Anna’s flag than Lupe’s?
2
3
1 4 yd2
Lesson
Main
Lesson
12-5
Feature
Exponents and Division
LESSON 12-5
Course 3
Check Skills You’ll Need
(For help, go to Lesson 2-7.)
1. Vocabulary Review Is the expression x5 an exponent or a power?
Write each expression using exponents.
2. 7 • 7 • 7 • 7
3. 4 • 4 • 4
4. 5 • 5
5. 1 • 1 • 1 • 1 • 1
Check Skills You’ll Need
Lesson
Main
Lesson
12-5
Feature
Exponents and Division
LESSON 12-5
Course 3
Check Skills You’ll Need
Solutions
1. power
2. 7 is a factor 4 times: 74
3. 4 is a factor 3 times: 43
4. 5 is a factor 2 times: 52
5. 1 is a factor 5 times: 15
Lesson
Main
Lesson
12-5
Feature
Exponents and Division
LESSON 12-5
Course 3
Additional Examples
14
Write x9 using a single exponent.
x
x14
(14 – 9)
=
x
9
x
= x5
Subtract exponents with the same base.
Simplify.
Quick Check
Lesson
Main
Lesson
12-5
Feature
Exponents and Division
LESSON 12-5
Course 3
Additional Examples
Quick Check
The distance between the sun and Jupiter is about 4.84 x 108
miles. Light travels at about 1.1 x 107 miles per minute. Use the formula
time = distance to estimate how long it takes sunlight to reach Jupiter.
speed
Write your answer in standard form.
time = distance
speed
Use the formula for time.
= 4.84 x
1.1
108
107
Substitute. Write as a product of
quotients.
= 4.84 x
1.1
4.4 x
101
Subtract exponents.
101
Divide.
It takes about 4.4 x 101 minutes, or 44 minutes, for
sunlight to reach Jupiter.
Lesson
Main
Lesson
12-5
Feature
Exponents and Division
LESSON 12-5
Course 3
Additional Examples
Simplify each expression.
a. (–5)0
(–5)0 = 1
Simplify.
b. 5y0
5y0 = 5 1 = 5
Simplify.
Quick Check
Lesson
Main
Lesson
12-5
Feature
Exponents and Division
LESSON 12-5
Course 3
Additional Examples
Simplify each expression.
a. 2–3
b. (p) –8
2–3 =
1
23
Write the expression using a
positive exponent.
=
1
8
Simplify.
1
p8
Quick Check
Lesson
Main
Lesson
12-5
Feature
Exponents and Division
LESSON 12-5
Course 3
Lesson Quiz
97
1. Write 94 using a single exponent.
93
Simplify each expression.
2. 60
1
3. 4–2
1
16
4. Jupiter’s diameter is about 1.43 x 105 kilometers. Earth’s diameter is
about 1.28 x 104 kilometers. How many times greater is Jupiter’s
diameter than Earth’s diameter? Round the answer to the nearest tenth.
11.2 times greater
Lesson
Main
Lesson
12-5
Feature