M3: Chapters 4 &12 Notes
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Academic
Chapters 4 & 12
Notes
Polynomials
Name____________________Pd.____
M3: Chapters 4 &12 Notes
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Chapters 4 and 12 Vocabulary
List of Words
Section 12.1:
 Monomial
 Binomial
 Trinomial
 Polynomial
 Term
 Degree of a term
 Degree of a polynomial
 Standard form
Section 4.7
 Scientific notation
M3: Chapters 4 &12 Notes
Page 3 of 22
Finding Rules of Exponents Activity
Learning Goal: Use patterns to discover rules for multiplying and dividing powers.
Products
Expression
Expression written as repeated
multiplication
Number of
factors
Product as a
power
2 4  23
(2  2  2  2)  (2  2  2)
7
27
31  34
(3)  (3  3  3  3)
Number of
factors
Product as a
power
5
25
52  54
Quotients
Expression
28
23
35
33
57
56
Expression written as repeated
multiplication
22222222
222
33333
333
1. In the Products Table, how are the exponents in the first and last columns related?
2. Use your answer for Question 1 to write
107 103 as a power.
3. In the Quotient Table, how are the exponents in the first and last columns related?
4. Use your answer to Question 3 to write the quotient
69
67
as a power.
M3: Chapters 4 &12 Notes
Page 4 of 22
Section 4.5: Rules of Exponents
Learning Goal: We will multiply and divide powers.
Multiplying Powers with the Same Base:
a 4  a 3  ( a  a  a  a)  ( a  a  a ) 
Example 1: Using the Product of Powers Property
Lake Powell, the reservoir behind the Glen Canyon Dam in Arizona, can
12
hold about 10 cubic feet of water when full. There are about
10 27 water molecules in 1 cubic foot of water. About how many water
molecules can the reservoir hold?
ON YOUR OWN:
Example 2: Using the Product of Powers Property
a.
b10  b8
b.
5m 2  6m 4
M3: Chapters 4 &12 Notes
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ON YOUR OWN:
Find the product. Write your answer using exponents.
Quotients of Powers:
a5 a  a  a  a  a


2
a
aa
Example 3: Using the Quotient of Powers Property
a.
58
53
12m5
b.
3m
c.
ON YOUR OWN:
Find the quotient. Write your answer using exponents.
7c 9
21c 3
M3: Chapters 4 &12 Notes
Example 4: Using Both Properties of Powers
Simplify
5x 4  x 6
10 x 5
ON YOUR OWN:
EXTRA PRACTICE:
Page 6 of 22
M3: Chapters 4 &12 Notes
Page 7 of 22
Section 4.6: Negative and Zero Exponents
Learning Goal: We will work with negative and zero exponents.
Pattern for Powers of 2:
23
22
21
20
2 1
2 2
What happens when you divide powers with the same base and get zero
as an exponent?
34

34
What happens when you divide powers with the same base and get a
negative exponent?
32

4
3
M3: Chapters 4 &12 Notes
Page 8 of 22
Example 1: Powers with Negative and Zero Exponents
Write the expression using only positive exponents.
a.
7 2
b.
a 5b0
c.
4m4 n3
ON YOUR OWN:
Example 2: Rewriting Fractions
Write the expression without using a fraction bar.
1
a.
16
a2
b. 3
c
c.
1
25
d.
x2
y3
ON YOUR OWN:
Write the expression without using a fraction bar.
***You can use the products of powers and quotient of powers
properties with problems involving negative exponents.***
M3: Chapters 4 &12 Notes
Page 9 of 22
Example 3: Using Powers Properties with Negative Exponents
Find the product or quotient. Write your answer using only positive
exponents.
a.
38  310
ON YOUR OWN:
8n 3
b.
n2
M3: Chapters 4 &12 Notes
Page 10 of 22
Section 4.7: Scientific Notation
Learning Goal:
 We will write and evaluate numbers in scientific notation.
 We will calculate with scientific notation.
Example 1: Writing Numbers in Scientific Notation
a. An earthquake of magnitude 8
b. A computer chip can be as
on the Richter scale releases
small as 0.0000005 meter long.
the equivalent energy of about
Write this number in scientific
5,643,000 metric tons of
notation.
explosive. Write this number in
scientific notation.
ON YOUR OWN:
Example 2: Writing Numbers in Standard Form
Write the number in standard form.
a.
1.85  10 6
b.
3.29  104
M3: Chapters 4 &12 Notes
Page 11 of 22
ON YOUR OWN:
**To compare numbers written in scientific notation, first compare
_______________________________, then compare the
_____________________________.
Example 3: Ordering Numbers Using Scientific Notation
ON YOUR OWN:
M3: Chapters 4 &12 Notes
Page 12 of 22
Example 4: Multiplying Numbers in Scientific Notation
6
The Great Pyramid of Giza in Egypt contains about 2.3  10 blocks of
3
stone. On the average, each block of stone weighs about 5  10 lb.
About how many pounds of stone does the Great Pyramid contain?
ON YOUR OWN:
Multiply. Express each result in scientific notation.
4
6
8
4
a. 4  10 6  10 
b. 7.1  10 8  10



M3: Chapters 4 &12 Notes
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Section 12.1: Polynomials
Learning Goal: We will classify and write polynomials in standard form.
Vocabulary:
 Polynomial – a sum of monomials (many terms)
 Term – the monomials that are added together in the polynomial
Monomial (1 term)
Binomial (2 terms)
Trinomial (3 terms)
Example 1: Identifying and Classifying Polynomials
Tell whether the expression is a polynomial. If it is a polynomial, list
its terms and classify it.
k
a. x 2
b. 2s 1  s
c. k 2  k 3 
4
ON YOUR OWN:
M3: Chapters 4 &12 Notes
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Vocabulary:
 Degree of a term – the sum of the exponents of the variables in
the term. (degree of a nonzero constant is 0)
 Degree of a polynomial – the greatest degree of the terms of the
polynomial
Example 2: Finding the Degree of a Polynomial
Find the degree of the polynomial.
a. 6c 2  5c  3
b. 2a 3  5a 4b  3b 6
ON YOUR OWN:
 Standard form – the form in which all like terms are combined
and the terms are arranged so that the degree of each term
decreases or stays the same from left to right
Example 3: Writing a Polynomial in Standard Form
Write 2m 3  3m  4m 5  m 3  m as a polynomial in standard form.
M3: Chapters 4 &12 Notes
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Write 3k 3  2k 2   k 4  2k 3 as a polynomial in standard form.
ON YOUR OWN:
Write the expression as a polynomial in standard form.
2
a. 3 x  4 x  8
3
3
b. 7t  2  t  5
Example 4: Evaluating a Polynomial
Evaluate the polynomial
ON YOUR OWN:
 4t 2  8t  1


c. 3 4b  b  b
when t  3 .
2
2
M3: Chapters 4 &12 Notes
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Section 12.2: Adding and Subtracting Polynomials
Learning Goal: We will add and subtract polynomials.
**You add polynomials by combining __________________________.
Example 1: Adding Polynomials Vertically
Find the sum.
a. (3 y 3  8 y  12)  ( y 3  6 y 2  3 y  4) b. (9b 4  b3  7b 2  17)  (12b3  4b 2  3b  1)
ON YOUR OWN:
Find the sum using a vertical format.
2
2
a. 5 p  3 p  7  2 p  8 p  4

 


 

3
2
b.  w  w  2  w  5
Example 2: Adding Polynomials Horizontally
The number of passengers waiting in Terminal A of an airport h hours after
the first scheduled flight of the day is given by the polynomial 350  80h .
The number of passengers waiting in Terminal B is given by the polynomial
220  75h .
a. Find the total number of
b. Find the total number of
passengers T waiting in both
passengers waiting in both
terminals after h hours.
terminals 4 hours after the first
scheduled flight of the day.
M3: Chapters 4 &12 Notes
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ON YOUR OWN:
SUBTRACTING POLYNOMIALS:
You can subtract a polynomial by ________________________.
To find the opposite of a polynomial, _____________________.
Example 3: Subtracting Polynomials Vertically
Find the difference (5 y 2  9 y  3)  ( y 2  2 y  6) .
ON YOUR OWN:
M3: Chapters 4 &12 Notes
Page 18 of 22
Example 4: Subtracting Polynomials Horizontally
Find the difference (7 g 2  2 g  5)  ( g 2  6 g  1) .
ON YOUR OWN:
Find the difference using a horizontal format.
a. 10 y 2  y  6  7 y 2  3 y  5
b. 5n 3  n  2  n 3  6n 2  9
EXTRA PRACTICE:
M3: Chapters 4 &12 Notes
Page 19 of 22
Section 12.3 Notes: Multiplying Monomials and Polynomials
Learning Goal: We will multiply polynomials and monomials.
Example 1: Multiplying a Monomial and a Binomial
Find the product.
a.
8 p3 p  4

2
3
b.  2 x 6 x  2 x

ON YOUR OWN:
Example 2: Multiplying Polynomials in Word Problems
Natalie is creating a dog pen attached to her house using 25 feet of
fencing. Write a polynomial expression in terms of l for the area of
the dog pen.
M3: Chapters 4 &12 Notes
Page 20 of 22
Example 3: Multiplying a Monomial and a Trinomial
Find the product.
a. ( y 4  7 y 3  2 y 2 )(3 y)
b. y 2 z( z 2  3 yz  2 y 2 )
ON YOUR OWN:
Find the product.

2
a. 3x 4 x  x  2



2
b.  8 y 2 y  3 y  5
DIVIDING BY A MONOMIAL:
To divide a polynomial by a monomial:
1.
2.
Example 4: Dividing a Polynomial by a Monomial
15 z 4  9 z 3  3 z 2
Find the quotient
.
3z 2
ON YOUR OWN:

2
2
c. a ab  3b  b

M3: Chapters 4 &12 Notes
Page 21 of 22
Section 12.4: Multiplying Binomials
Learning Goal: We will multiply binomials.
Example 1: Multiplying Binomials Using a Table
Find the product (c  3)(4c  1) .
ON YOUR OWN:
Use a table to find the product.
a. 4 x  12 x  3
b. 2a  35a  6
c. 6b  9 2b  5
Example 2: Using the Distributive Property
A rectangular pool has a length that is three times its width. You plan
to build a deck around the pool that is 3 feet wide. Write a polynomial
expression for the combined area of the pool’s surface and the deck
around it.
M3: Chapters 4 &12 Notes
Page 22 of 22
FOIL METHOD:
F–
O–
(2 x  5)(6 x  1) 
I–
L–
Example 3: Using the FOIL Method
Find the product (7a  3)(2a  9) .
ON YOUR OWN:
Find the product.
a. x  5x  9
b. z  610 z  2
c. m  3m  2