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Integrated Algebra
Chapter 7: Exponents and Polynomials
Name:______________________________
Teacher:____________________________
Pd: _______
2
Table of Contents
o Chapter 7-1: SWBAT: Evaluate and simplify expressions containing zero and integer exponents.
Pgs: 3-5
HW: Page 6 (Evens Only)
o Chapter 7-2: SWBAT: Convert between standard notation and scientific notation.
Pgs: 7-10
HW: Page 11
o Lesson 7-3: SWBAT: Use Multiplication Properties of Exponents to Evaluate and Simplify
Expressions.
Pgs: 12-15
HW: Page 16
o Lesson 7-4: SWBAT: Use Division Properties of Exponents to Evaluate and Simplify Expressions
Pgs: 17-19
HW: Page 20
o Half Period Quiz Lessons: 7-1 to7-4
o Lesson 12-6: SWBAT: Divide a polynomial by a monomial.
Pgs: 21-22
HW: Page 23
o Lesson 7-6: SWBAT: Add and Subtract Polynomials.
Pgs: 24-26
HW: Page 27
o Lesson 7-7: SWBAT: Multiply Polynomials by a monomial.
Pgs: 28-31
HW: Page 32
o Lesson 7-7: SWBAT: Multiply two binomials.
Pgs: 33-36
HW: Page 37
o Full Period Quiz Lessons: 12-6, 7-5 to7-7
o Practice Test: Review using E-Clickers
Pgs: 38-40
3
Chapter 7 – 1
SWBAT: Evaluate and simplify expressions containing zero and integer exponents.
Warm – Up
Evaluate each expression for the given values of the variables.
You have seen positive exponents. Recall that to simplify 32, use 3 as a factor 2 times: 3
2 = 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.
Example 1: Simplify these expressions.
Practice: Simplify these expressions.
1. x3y2 for x = –1 and y = 10 2. y2
x3 for x = –1 and y = 10
a. 2–4
b. 70
c. (-5)–3
d. -5–3
a. 3–3 c. (-2)–3
b. 150 d. -2–3
4
Example 2: Evaluate the expression for the given value of the variables.
a. x–2 for x = 4
Practice: Evaluate the expression for the given value of the variables.
a. p–3 for p = 4
Example 3: Simplifying Expressions with Zero and Negative Numbers
a. 7w–4
Practice: Simplifying Expressions with Zero and Negative Numbers
a. 2r0m-3
b. –2a0b-4 for a = 5 and b = –3
b. 8a-2b0 for a = -2 and b = 6
b. -5
k–2c.
a0b-2
c-3d6
b. r–3
7c.
g4
h-6
d. 2a-5
b-6e.
x-5
3y12 f. 20p-1
5q-3
5
Challenge Problem: Simplify
[2-2 + (6 + 2)0]
Summary:
Exit Ticket:
6
Chapter 7-1 Homework (EVENS ONLY)
7
Chapter 7 – 2
SWBAT: Convert between standard notation and scientific notation.
Warm – Up
Simplify:
a3c-4x5
a3b-2
c-1
Scientific Notation
…
…
…
(1)
(2)
(3)
3.2 x 1013? 23.6 x 10-8?
8
Example 1: Scientific Notation to Standard Form
The exponent represents the number of places the decimal needs to be moved.
Move decimal point: to the ________ for ____________ exponents of 10
OR to the ________ for ____________ exponents of 10.
1.015 x 10-8
5.024 x 103
Practice Problems: Express in standard form:
1) 5.02 x 103 3) 603 x 10-4
2) 52.8 x 106 4) 0.03 x 10-2
Example 2:Standard Form to Scientific Notation
Remember, the decimal is at the end of the final zero.
The decimal must be moved to ensure that the _____________ is between _____________.
How many places did it move? This number will be the _____________.
Express in scientific notation:
4,750,000
0.000789
Practice Problems: Express in scientific notation:
1) 4500 3) 6,560,000
2) 0.00002 4) 0.00203
9
Example 3:Using Scientific Notation to compute products and quotients of numbers.
Multiplying and Scientific Notation
Example:
To find the answer to this problem, we can multiply 9.3 and 6.2 and get 57.66.
We can also multiply to get .
So putting these two pieces together we get:
But remember that to be in scientific notation…
_____________________________________________________________________________
1) What is the product of and written in scientific notation?
1) 2) 3) 4)
2) What is the product of , , and expressed in scientific notation?
1) 2) 3) 4)
Dividing and Scientific Notation
Example:
3) What is the quotient of and ?
1) 2) 3) 4)
4) The quotient of and expressed in scientific notation is
1) 4,000
2) 40,000
3) 4)
10
Challenge Problem:
Summary:
Exit Ticket:
11
Chapter 7-2 Homework
Part A: Express each of the following in standard form.
1) 5.2 x 103 2) 3.6 x 10
1 3) 9.65 x 10
–4
4) 6.452 x 102 5) 8.5 x 10
–2 6) 8.77 x 10
–1
Part B: Express each of the following in scientific notation.
7) 78,000 8) 16 9) 0.00053
10) 0.0043 11) 250000000000 12) 0.875
Part C: Multiplying and Dividing Scientific Notation
13) (6.02 x 1023
) (8.65 x 104) 14) (3.2 x 10
–7) (8.6 x 10
9) 15) (5.4 x 10
4) (2.2 x 10
7)
16) (1.6 x 105) (2.4 x 10
15) 17) (7.0 x 10
28) (–3.2 x 10
–20) (–6.4 x 10
35)
18) If is divided by , the result is
1) 1
2) 0.01
3) 4)
12
Chapter 7 – 3
SWBAT: Use Multiplication Properties of Exponents to Evaluate and Simplify Expressions.
Warm – Up Use a calculator to find the value of each expression.
a. 53 55 =
c. 51 57 =
b. 56 52 =
d. 54 54 =
Example 1: Products of Powers
Notice that when we multiply powers with the same base, you ___________ the exponents.
For any nonzero number a and any integer m and n, ______________________.
Simplify.
a) c4 d3 c2
b) 5x 2y4 3x8
Rearrange factors.
Multiply coefficients.
Add exponents of powers
with the same base.
Solution
13
Practice: Simplify.
Example 2: Raise a Power to a Power Simplify.
1) (x3)4
Notice that when we raise a power to a power, you ____________ the exponents.
For any nonzero number a and any integer m and n, ______________________.
Practice: Simplify.
Example 3: Raise a Product to a Power
We can use repeated multiplication to simplify expressions like (5y)³.
Simplify.
Method 1 Method 2
(5y)³ (5y)³
14
Notice that when we raise a product to a power, you _____________________________ and ______________.
For any nonzero, real numbers a and b and any integer n, ___________________.
Practice: Simplify.
Additional Examples: Simplify.
15
Challenge Problem
Summary
Exit Ticket
16
Chapter 7-3 Homework:
Simplify.
17
Chapter 7 – 4
SWBAT: Use Division Properties of Exponents to Evaluate and Simplify Expressions
Warm – Up Simplify.
4(3c3)3
Example 1: Dividing Powers with the Same Base
When we divide powers with the same base, you ______________ the exponents.
For any number a and any integers m and n, ______________________.
Simplify. Use only positive exponents.
Practice: Simplify. Use only positive exponents.
1)
45
5 2)
4
3
4
3
x
x 3)
6 3
3
a b
ab
4)
10
2
z
z 5)
9
3
x
x 6)
8 4
3 12 3
r t
r s t
7)
3 5
6 3
4
2
x y
x y 8) 3
5
15
ab
a
9)
4 2 3
5 2
18
3
a b c
a bc
x6
x2
x2
x6
a4b-2
a2b3
10x3y5
10x5y2
18
Example 2: Finding Positive Powers of a Quotient
For any nonzero real numbers a and b and positive integer n, ______________________.
Simplify. Use only positive exponents.
3
5 4
Practice: Simplify. Use only positive exponents.
Example 3: Finding Negative Powers of a Quotient
For any nonzero real numbers a and b and positive integer n, ______________________.
Simplify. Use only positive exponents.
2
3 -4
Practice: Simplify. Use only positive exponents.
19
Challenge Problem: Simplify. Use only positive exponents.
Summary
Exit Ticket
20
Chapter 7-4 Homework:
Simplify the following using the rules of exponents.
1)
12
4
h
h 2)
10
6
c
c 3)
5
4
3
2
x
x
4)
15 6
5 8 3
r t
r s t 5)
9 4
9
a b
a b 6) 2
39
z
xz
7) 32
23
7
28
qp
qp 8) 3
614
x
yx 9)
413
9320
yx
zyx
21
Chapter 12 – 6
SWBAT: Divide a polynomial by a monomial.
Warm – Up
****Half Period Quiz****
Example Divide.
(6x3 + 8x2 - 4x) 2x
Practice Problems Divide.
(8x3 - 4x2 + 12x) (-4x2)(15x3 - 20x2 + 5x) 5x
22
Challenge Problem:
Divide and Simplify.
Summary:
Exit Ticket:
When is divided by , the quotient is
1)
2)
3)
4)
45a4b3 - 90a3b
15a2b
23
Chapter 12 – 6 Homework Directions: Simplify each expression.
1)
x
xxx 234 63 2)
2
256
8
8408
x
xxx
3)
2
234
9
18369
x
xxx 4)
m
mmm
2
222 234
5)
x
xxx
5
2550100 58
6)
x
xxx
2
10412 23
7)
2
24
10
1030
r
rr 8)
2
34
4
1648
x
xx
9) xxxx 6)61236( 23 10) * 52 3)3612( xxx
24
SWBAT: Add and Subtract Polynomials.
Chapter 7 – 6
Warm – Up
Subtract )8( 2aa from )834( 2aa
Remember!
___________________________ are:
- constants
- terms with the SAME variable(s) raised to the SAME powers.
Example 1: Adding Polynomials
Practice: Adding Polynomials
1. 2.
3. )227()12( 22 xxx 4. )76()48( 22 nnnn
5. )32()33( 232 mmm 6. )25()227( 334 bbb
Find (2x2 - 3x + 4) + (3x2 + 2x - 3).
25
7.
Example 2: Subtracting Polynomials
Practice Problems: Subtracting Polynomials
Find each difference. Write your answer in standard form.
8. 9.
10. 11.
12. )46()354( 22 bbb 13.
Find (7x2 - 3x + 1) - (x2 + 4x - 2).
a. (3x3 + 5x - 2) - (x3 - 4x - 3) b. (2a3 + 9a - 2) - (a2 + 4a - 7)
c. (y2 - 7y) - (-4y2 + 3y - 1) d. (x2 + 2x + 5) - (2x2 - 4x - 5)
26
Challenge Problems Find the perimeter of the figure below.
Summary
Exit Ticket
What is the sum of and ?
1)
2)
3)
4)
5x + 2
6x - 10
27
Chapter 7-6 - Homework:
Add or subtract.
From )462( 2bb subtract )554( 2 bb
Find the difference when )942( 2 xx is subtracted from )185( 2 xx
28
Chapter 7 – 7 (Day 1) SWBAT: Multiply polynomials by a monomial.
Warm – Up Use the distributive property to simplify each expression.
a. 3(x + 2)
b. 2(x + y)
c. (m - 7)2
d. -(b - c)
Example 1: Product of Monomials Step 1: Group factors with like bases together.
Step 2: Multiply.
Multiply 3x2 and 2x3
Multiply 4a2b3 and 3a3b
Practice Problems: Product of Monomials
1. 2. 3.
Example 2: Product of a Polynomial and a Monomial Step 1: Distribute.
Step 2: Group factors with like bases together.
Step 3: Multiply.
Multiply 3x and (2x + 1)
Multiply 2x2y and (3x - y)
29
Practice Problems: Product of a Polynomial and a Monomial
Practice Find the product.
1) )5(3 x 2) 7( 5 8)x
3) 2( 2)x x 4) 2 ( 12)k k
5) 2 3(3 )a a a 6)
2 53 (9 4 )x x x
7) 4 62 ( 3 )y y y 8)
2 9 25 (2 3 )s s s
9) )142(3 2 xx 10) )422(5 2 xxx
Example 3: Product of a Polynomial and a Monomial Step 1: Distribute.
Step 2: Group factors with like bases together.
Step 3: Multiply.
30
Practice Problems: Product of a Polynomial and a Monomial Step 1: Distribute.
Step 2: Group factors with like bases together.
Step 3: Multiply.
1) 2 6 2 98 (2 4 ) 3 ( 2 9 )x y x y xy x y 2) 2 25 (4 2 ) 3 ( 2 4 )a a a a a a
3) 2 22 (5 6) 5 ( 3 4) 7( 5)a a a a a a 4) 2 25 ( 7 3) 2 (2 19 2 )w w w w w w
5) 26 (2 3) 5(2 9 3)t t t t 6) 3 4 2 2 38 ( 2 ) 4 (1 6 )b b b c b b b c b
7) 2 2 3 34 ( 8 3 ) 7 (8 1)x x x x x xy 8)
3 2 4 5 5 7 88 (3 2 ) 2 (5 7 6)x y xy x y x x y xy
31
Challenge Problem:
Summary:
Exit Ticket:
32
Chapter 7-7 (Day 1) Homework: Find the product.
33
Chapter 7 – 7 (Day 2)
SWBAT: Multiply two binomials.
Warm – Up
1) )12(4 2 xx 2) )32(3 22 baab 3) )3(2 223 abbaba
Method 1: Use the Distributive Property Step 1: Distribute the first polynomial over the terms of the second. Step 2: Multiply.
Step 3: Combine like terms.
(2x + 1)(x - 5)
Method 2: Use a Box Step 1: Write the product of the monomials in each row or column.
Step 2: Add the terms inside the rectangle.
(2x + 1)(x - 5)
(2x + 1)(x - 5)
34
Method 3: Use FOIL Step 1: Multiply the FIRST terms.
Step 2: Multiply the OUTER terms.
Step 3: Multiply the INNER terms.
Step 4: Multiply the LAST terms.
Step 5: Combine like terms.
First
Outer
Inner
Last
Practice Problems Directions: Simplify each expression by using the distribute property, box method or FOIL method.
1) ( 9)( 9)b b 2) 2)2( x
3) (3 4)(4 7)x x 4) (2 3)( 4)x x
5) The length of a rectangle is 53 x and its width is 2x . Express the area of the rectangle.
(2x + 1)(x - 5)
35
Example 2: Multiplying Polynomials
Multiply.
)4)(32( yxyx 2 2 2 2( )( )c d c d
Practice Problems: Multiplying Polynomials
Multiply.
1) (4 )(5 2 )x y x y 2) )6)(2( 2233 yxyx
3) )4)(2( 22 baba 4) )5)(2( 2222 yxyx
5) (2x3 – 5)
2 6) (x
3 – y
5)
2
36
Challenge Problem
Multiply and Simplify.
)3()52( 2 xx 22 )15()93( xx
Summary
Exit Ticket
37
Chapter 7-7 (Day 2) Homework
Find the product.
The length of a rectangle is 2 5x and its width is 7x . Express the area of the rectangle.
38
Chapter 7 Review SWBAT: Assess their mastery of concepts and skills.
7-2: Scientific Notation
1. Which is the following is the standard form of 4 x 10-6?
A. 0.000004 B. 0.0000004 C. 40,000 D. 400,000
2. In 2000, the population of Thomasville, North Carolina, was about 20,000 people. Write this population in scientific notation.
A. 0.2 x 105 B. 2 x 104 C. 2 x 105 D. 20 x 103
7-3: Multiplication Properties of Exponents
3. Simplify 86 ∙ 83 . A. 82 B. 83 C. 89 D. 818
4. Simplify (a3b)2.
A. a3b2 B. a6b C. a6b2 D. a9b2
5. Simplify (23)3.
A. 20 B. 21 C. 26 D. 29
6. Which of the following is equivalent to 6-4? A. (-6)4 B. 1
64 C. -24 D. 24
7. Evaluate x-3 for x = 2. A. -8 B. -6 C. 1
9 D. 1
8
8. Evaluate 4w2 for x = 2. A. 103 B. 112 C. 403 D. 529
7-4: Division Properties of Exponents
9. Simplify 310 . 32
A. 35 B. 38 C. 312 D. 320
10. Simplify . A. 9
7 B. 6
14 C. 9
49 D. 7 9
39
11. Simplify A. -25
81 B. 81
25 C. 10
18 D. 18
10
12. Simplify . A. B. C. D. 81x12
12-6: Dividing Polynomials
13. Divide (12x2 + 6x – 3) ÷ 3. A. 12x2 + 6x - 1 B. 12x2 + 6x C. 4x2 + 2x - 1 D. 12x2 + 2x - 1
7-5: Describing Polynomials
14. Which polynomial is written in standard form? A. -5x3 + 2x + 9x2 B. -5x3 + 9x2 + 2x C. 2x + 9x2 - 5x3 D. 9x2 + 2x - 5x3
7-6: Adding Polynomials
15. Add (2x3 – 5) + (x3+ 3). A. 3x3 - 2 B. 3x6 - 2 C. 3x6 - 15 D. 3x3 + 8
7-6: Subtracting Polynomials
16. Subtract (6a3 + 3a) - (4a2 + 2a). A. 2a2 + a B. 2a2 + 5a C. 3 D. 3a3
17. Subtract (7x + 2) - (x - 4). A. 6x - 6 B. 6x + 6 C. 6x2 + 6 D. 6x2 - 6
3x20
x2
3x20
x8
81x20
x8
40
7-7: Multiplication by a Monomial
18. Multiply (4r3)(2r5). A. 8r8 B. 8r15 C. 2048r8 D. 2048r15
19. Multiply (3x3)(2y)2(4x4).
A. 48x12y2 B. 28x12y2 C. 28x7y2 D. 48x7y2
7-7: Multiplying Binomial 22. Multiply (b + 3)(b2 + 2b).
A. b3 + 2b3 + 3b B. b3 + 5b2 + 6b C. b3 + 6b2 + 3b D. 4b5 + 8b3 + 3b
23. Multiply (x - 4)(x2 + 9x).
A. 2x2 + 5x2 – 39x B. x3 + 5x2 – 36x C. 2x3 + 5x2 + 33x D. x3 + 5x2 – 39x
7-7: Multiplying two Binomials
20. Multiply (x + 2)(x + 3). A. x2 + 6 B. x2 + 5x + 5 C. 2x + 5 D. x2 + 5x + 6
21. Multiply (2x - 3)2.
A. 2x2 - 3 B. 2x2 - 12x + 9 C. 4x2 - 3 D. 4x2 - 12x + 9
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