SOUTHEAST MIDDLE SCHOOL
Horrel Hill Road, Hopkins, South Carolina
Name:__________________________Period:____________Date:______________
Algebra 1
Day 1 and 2
Lesson Plan: Recalling the Laws of Exponents
8.EEI.1 Understand and apply the laws of exponents (i.e., product
Standards
rule, quotient
rule, power to a power, product to a power, quotient to a power,
zero power
property, negative exponents) to simplify numerical expressions that
include
integer exponents.
Learning Targets I can apply the laws of exponents.
I Can Statements
Essential
Question(s)
How can the laws of exponents be applied in real-world situations?
Resources
Learning
Activities or
Experiences
1|P a g e
You will need a pair of scissors and a glue stick to complete this
assignment. All
answers should be written on the page provided.
1. Complete at least 3 topics of your ALEKS pathway. (if available)
Review attached notes and complete the “Exponent Rules
2. Puzzle.”
Practice
3. Assessment
Dr. Ramirez Algebra 1 Class
Lesson Notes
Day 1: Exponent Rules Puzzle
1. Cut out the nine puzzle pieces.
2. Pair up the matching expressions
(each non-simplified expression has a
matching simplified expression).
3. When complete, the puzzle will be a three-by-three square. Glue
your final arrangement on the page provided. GOOD LUCK!
2|P a g e
Dr. Ramirez Algebra 1 Class
3|P a g e
Dr. Ramirez Algebra 1 Class
Exponent Rules Puzzle Solution
4|P a g e
Dr. Ramirez Algebra 1 Class
Day 2 : Assessment
5|P a g e
Dr. Ramirez Algebra 1 Class
6|P a g e
Dr. Ramirez Algebra 1 Class
SOUTHEAST MIDDLE SCHOOL
Horrel Hill Road, Hopkins, South Carolina
Algebra 1
Day 3 and 4
Lesson Plan: Introduction to Polynomials
Standards
A1.ASE.1* Interpret the meanings of coefficients, factors, terms, and expressions
based on their real-world contexts. Interpret complicated expressions as being
composed of simpler expressions. (Limit to linear; quadratic; exponential.)
A1.ASE.2* Analyze the structure of binomials, trinomials, and other polynomials
in order to rewrite equivalent expressions A1.AAPR.1* Add, subtract, and
multiply polynomials and understand that polynomials are closed under these
operations. (Limit to linear; quadratic.
A1.AAPR.1* Add, subtract, and multiply polynomials and understand that
polynomials are closed under these operations. (Limit to linear; quadratic
I can add, subtract, and combine polynomial expressions.
I can apply polynomial addition and subtract to contextual problems
Learning Targets
I Can Statements
Essential
Question(s)
How does the degree of a polynomial change its graph? How do we perform
operations on polynomials? How do we factor polynomials?
Khan Academy Adding and Subtracting Polynomials
https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:polyarithmetic/x2ec2f6f830c9fb89:poly-add-sub/v/adding-and-subtractingpolynomials-1
MathBitsNotesbook.com Polynomials: Add and Subtract Mathematics
https://mathbitsnotebook.com/Algebra1/Polynomials/POaddsubtract.html
Resources
Learning Activities
or
Experiences
Powerpoint presentation : All about Polynomials
1. Complete at least 3 topics of your ALEKS pathway if
available)
2. Read the Lecture entitled “All About Polynomials”
3. Do the Activity
4. Answer the Assessment
Day 3 Activity
Fill out this table with what is needed in order to understand the different concepts of polynomials. You
may use the powerpoint. Just give one example and one counterexample (not an example).
No.
TERM
1
Polynomial
2
Term
3
Coefficient
4
Constant
6
Monomial
DEFINITION
EXAMPLE
COUNTEREXAMPLE
A polynomial that has only
x2
x+ y
one terms
7
Binomial
8
Trinomial
9
Multinomial
10
11
Degree of a
Polynomial
Degree of a Term
12
Like Terms
13
Unlike Terms
14
Linear Polynomial
Polynomials with the
highest degree is 1
+1
15 Quadratic Polynomial Polynomials with the
highest degree is 2
16
Cubic Polynomial
Polynomials with the
highest degree is 3
x+ y
+
–
x3+ y
+
x4+ y
Day 4 Activity
Terms Involving Polynomials
Multiple Choice
Identify the choice that best completes the statement or answers the question.
1. Find the degree of the monomial
a.
–5
b.
11
2. Find the degree of the polynomial
a.
14
b.
6
.
c.
d.
7
4
c.
d.
12
9
.
3. Write the polynomial 3x2 – 8x – 12x5 – 5x3 + 2x4 – 6 in standard form. Then give the leading coefficient.
a.
The leading coefficient is –6.
b.
The leading coefficient is –6.
c.
The leading coefficient is –12.
d.
The leading coefficient is –12.
4. ____Classify the polynomial according to its degree and number of terms.
a. The polynomial is a linear monomial.
b. The polynomial is a seventh degree monomial.
c. The polynomial is a quadratic binomial.
d. The polynomial is a seventh degree binomial.
5. Classify the polynomial according to its degree and number of terms.
–
+
a. The polynomial is a quartic monomial.
b. The polynomial is a linear trinomial.
c. The polynomial is a cubic trinomial.
d. The polynomial is a linear monomial.
6. Simplify by combining like terms:
4x2 - 3x + 11 -2x
a)4x2 - 5x + 11
b)11x + 11
c)16x2 - 5x + 11
d)10x2
7. Simplify this polynomial:3x2 - x + 2 - 5x2 + 8x - 5
a)8x2 + 9x + 7
b)-2x2 + 7x – 3
c)2x2 + 7x – 3
d)5x2 - 3
8. Classify by number of terms: 3x3 – 6x
a)Monomial
b)Binomial
d.) 4-Term Polynomial
c)Trinomial
9. How many terms are in this polynomial?
a)7
b)1
c)4
2x3 - 8x2 + 3x - 7
d)2
10. 3.What is the degree of this polynomial?2x4 - 3x5 + x
a)-3
b)2
c)4
d)5
11. 4.What is the constant of this polynomial?2x3 - 8x2 + 3x - 7
a)2
b)-8
c)3
d)-7
12. 5.What is the coefficient of this term?9x3
a)None
b)9
c)3
d)x
13. 6.Classify by number of terms:5x2 – 6x + 3
a)Monomial
b)Binomial
c)Trinomial
d)4-Term Polynomial
14. 7.Simplify the Expression:3x + 6 - 2x +7
a)14
b)14x
c)x + 13
d)x – 13
15. 8.What is the the DEGREE:4x3 - 5x2 + 2x - 1
a)1
b)2
c)3
d)4
16. 9.Classify:
a)Monomial
d)Polynomial
2n³
b)Binomial
17. What are like terms?
Terms that have....
a)the same variables
c)the same coefficients
c)Trinomia
b)the same exponents
d)the same variables and exponents
18. The number in front of the variable is called the....
a)exponent
b)base
c)coefficient
d)fronter
19. 12.Which pair is an example of like terms?
a)2a and 2b
b)x and x2
c)3k and 3k3
d)y2 and 2y2
20. Simplify this polynomial:n - 10 + 9n - 3
a)-3
b)9n – 3
c)n
d)10n – 13
SOUTHEAST MIDDLE SCHOOL
Horrel Hill Road, Hopkins, South Carolina
Algebra 1
Day 5 and 6
Lesson Plan: Introduction to Polynomials
Standards
A1.ASE.1* Interpret the meanings of coefficients, factors, terms, and expressions
based on their real-world contexts. Interpret complicated expressions as being
composed of simpler expressions. (Limit to linear; quadratic; exponential.)
A1.ASE.2* Analyze the structure of binomials, trinomials, and other polynomials
in order to rewrite equivalent expressions A1.AAPR.1* Add, subtract, and
multiply polynomials and understand that polynomials are closed under these
operations. (Limit to linear; quadratic.
A1.AAPR.1* Add, subtract, and multiply polynomials and understand that
polynomials are closed under these operations. (Limit to linear; quadratic
I can Add, subtract, and multiply polynomials and understand that polynomials
are closed under these operations.
I can apply polynomial addition and subtract to contextual problems
Learning Targets
I Can Statements
Essential
Question(s)
Resources
Learning Activities
or
Experiences
How does the degree of a polynomial change its graph? How do we perform
operations on polynomials? How do we factor polynomials?
Khan Academy Adding and Subtracting Polynomials
https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:polyarithmetic/x2ec2f6f830c9fb89:poly-add-sub/v/adding-and-subtractingpolynomials-1
MathBitsNotesbook.com Polynomials: Add and Subtract Mathematics
https://mathbitsnotebook.com/Algebra1/Polynomials/POaddsubtract.html
1. Complete at least 3 topics of your ALEKS pathway if
available)
2. Do Now
3. Read the Polynomial Lecture in Power point Presentation
4. Do the Activity
5. Answer the Assessment
Day 5 Activity
Students solve the numbered problems and find the matching lettered answer. They put the corresponding
letter into the table at the bottom of the page.
Day 6 Activity
Read and solve the problem. You may write on this test.
Circle your response on the accompany scantron.
Use the expression
1.
What is the degree of the expression?
A.
2.
4.
5.
6.
2
B.
3
C.
4
D.
5
15
D.
2
What is the leading coefficient of the expression?
A.
3.
for problems 1 and 2.
-4
B.
3
C.
Find the sum.
A.
C.
B.
D.
Subtract
A.
C.
B.
D.
Simplify
A.
C.
B.
D.
Simplify.
A.
-5x – 2
C.
B.
6x – 2
D.
6x – 6
Match the polynomial?
7.
10.
2d
A.
Trinomial
8.
B.
Monomial
9.
C.
Binomial
Write in stanard form:
.
_____
Exponents
Power
5
exponent
3
base
Example: 125  53 means that 53 is the exponential
form of the number 125.
53 means 3 factors of 5 or 5 x 5 x 5
The Laws of Exponents:
#1: Exponential form: The exponent of a power indicates
how many times the base multiplies itself.
x  x  x  x x  x  x  x
n
n times
n factors of x
Example: 5  5  5  5
3
#2: Multiplying Powers:
If you are multiplying Powers
with the same base, KEEP the BASE & ADD the EXPONENTS!
x x  x
m
So, I get it!
When you
multiply
Powers, you
add the
exponents!
n
m n
2 6  23  2 6  3  29
 512
#3: Dividing Powers: When dividing Powers with the
same base, KEEP the BASE & SUBTRACT the EXPONENTS!
m
x
m
n
mn

x

x

x
n
x
So, I get it!
When you
divide
Powers, you
subtract the
exponents!
6
2
6 2
4

2

2
2
2
 16
Try these:
12
1. 3  3 
2
2
7.
2. 52  54 
3.
8.
a a 
5
2
4. 2s  4s 
2
7
12 8
9.
5. (3)  (3) 
2
6.
3
s t s t 
2 4
7 3
s

4
s
9
3

5
3
s t

4 4
st
5 8
10.
36a b

4 5
4a b
SOLUTIONS
2
2 2
4
a a  a
5 2
a
1. 3  3  3  3  81
2 4
6
2
4
2. 5  5  5  5
2
3.
5
2
4. 2s  4s  2  4  s
2
7
5. (3)  (3)  (3)
2
6.
3
s t s t 
2 4
7 3
s
7
2 7
23
 8s
 (3)  243
2 7 43
t
9
5
s t
9 7
SOLUTIONS
12
7.
8.
9.
10.
s
12  4
8
s

s

4
s
9
3
9 5
4
3

3

81

5
3
12 8
s t
12  4 8 4
8 4
s
t

s
t

4 4
st
5 8
36a b
5  4 8 5
3
36

4

a
b

9
ab

4 5
4a b
#4: Power of a Power: If you are raising a Power to an
exponent, you multiply the exponents!
x 
n
m
So, when I
take a Power
to a power, I
multiply the
exponents
x
x 
3 5
=
mn
x
35
=
15
x
#5: Product Law of Exponents: If the product of the
bases is powered by the same exponent, then the result is a
multiplication of individual factors of the product, each powered
by the given exponent.
 xy 
So, when I take
a Power of a
Product, I apply
the exponent to
all factors of
the product.
n
x y
n
n
(ab)  a b
2
2
2
#6: Quotient Law of Exponents: If the quotient of the
bases is powered by the same exponent, then the result is both
numerator and denominator , each powered by the given exponent.
n
 x
x
   n
y
 y
So, when I take a
Power of a
Quotient, I apply
the exponent to
all parts of the
quotient.
n
4
 2  2 16
   4 
81
3 3
4
Try these:
 
1. 3
5
2 5

 
3. 2a  
4. 2 a b 
2. a
s
7.   
t 2
 39 
8.  5  
3 
3 4
2 3
2
5 3 2
5. (3a ) 
2 2
 
2 4 3
6. s t


2
 st 
9.  4  
 rt 
5 8 2
 36a b 
 
10. 
4 5 
 4a b 
8
SOLUTIONS
 
1. 3
2 5
 
2. a
3 4

10
3

a12
 
 2 a


3. 2 a
2
2 3
3
5 3 2
4. 2 a b
23
 8a
6
 222 a 52b32  24 a10 b 6  16a10 b 6
5. (3a )  3  a 22  9a 4
2
2 2
 
2 4 3
6. s t
23 43
s t
s t
6 12
SOLUTIONS
5
s
7.   
t
5
s
5
t
2
3 
8.  5   34 2  38
3 
9
2
4 2
2 8
 st 


st
s
t
9.  4   
  2

r
 rt 
 r 
8
 36 a b
10 
4 5
 4a b
5 8
2

  9ab 3



2
2 32
9 a b
2
 81a b
2 6
#7: Negative Law of Exponents: If the base is powered
by the negative exponent, then the base becomes reciprocal with the
positive exponent.
So, when I have a
Negative Exponent, I
switch the base to its
reciprocal with a
Positive Exponent.
Ha Ha!
If the base with the
negative exponent is in
the denominator, it
moves to the
numerator to lose its
negative sign!
x
m
1
 m
x
1
1
5  3 
5
125
and
3
1
2

3
9
2
3
#8: Zero Law of Exponents: Any base powered by zero
exponent equals one.
x 1
0
So zero
factors of a
base equals 1.
That makes
sense! Every
power has a
coefficient
of 1.
50  1
and
a0  1
and
(5a ) 0  1
Try these:
1.
2a b 
0
2

2.
y 2  y 4 
3.
a 
5 1
2

4. s  4s 

7
 
s t  
2
5. 3 x y
6.
3 4
2 4 0
1
2 
7.   
 x 2
 39 
8.  5  
3 
2
2
s t 
9.  4 4  
s t 
2
5
 36a 
10.  4 5  
 4a b 
2 2
SOLUTIONS


0
1. 2 a b  1
2
1
3. a   5
a
5
2
7
4. s  4s  4s
5 1

2
5. 3 x y
 
2 4 0
6. s t

3 4

4
 3 x y
 1
8
12

8
x

81 y12
SOLUTIONS
1
2 
7.  
 x 
9 2
3 
8.  5 
3 
2
1
x
4
  
4
 x
 3
2

4 2
1
3  8
3
8
s t 
 2  2 2
4 4
9.  4 4   s t   s t
s t 
10
2
5
b

2

2
10
 36a 
9
a
b

2


10.  4 5  
81a
4
a
b


2 2
Let’s Define it.
P O LY N O M I A L V O C A B U L A R Y
Term – a number or a product of a number and variables
raised to powers
Coefficient – numerical factor of a term
Constant – term which is only a number
Polynomial is a sum of terms involving variables raised to
a whole number exponent, with no variables appearing
in any denominator.
P O LY N O M I A L V O C A B U L A R Y
In the polynomial 7x5 + x2y2 – 4xy + 7
There are 4 terms: 7x5, x2y2, -4xy and 7.
The coefficient of term 7x5 is 7,
of term x2y2 is 1,
of term –4xy is –4 and
of term 7 is 7.
7 is a constant term.
T Y P E S O F P O LY N O M I A L S
Monomial is a polynomial with 1 term.
Binomial is a polynomial with 2 terms.
Trinomial is a polynomial with 3 terms.
Multinomial is a polynomial with 4 or more terms.
DEGREES
Degree of a term
To find the degree, take the sum of the exponents on the
variables contained in the term.
Degree of a constant is 0.
Degree of the term 5a4b3c is 8 (remember that c can be
written as c1).
Degree of a polynomial
To find the degree, take the largest degree of any term of the
polynomial.
Degree of 9x3 – 4x2 + 7 is 3.
E VA L U AT I N G P O LY N O M I A L S
Evaluating a polynomial for a particular value involves
replacing the value for the variable(s) involved.
Example
Find the value of 2x3 – 3x + 4 when x = 2.
2x3 – 3x + 4 = 2( 2)3 – 3( 2) + 4
= 2( 8) + 6 + 4
= 6
COMBINING LIKE TERMS
Like terms are terms that contain exactly the same variables raised
to exactly the same powers.
Warning!
Only like terms can be combined through addition and
subtraction.
Example
Combine like terms to simplify.
x2y + xy – y + 10x2y – 2y + xy
= x2y + 10x2y + xy + xy – y – 2y
(Like terms are grouped together)
= (1 + 10)x2y + (1 + 1)xy + (– 1 – 2)y = 11x2y + 2xy – 3y
ADDING AND SUBTRACTING
POLYNOMIALS
Let’s Add and Subtract!
ADDING AND SUBTRACTING POLYNOMIALS
Adding Polynomials
Combine all the like terms.
Subtracting Polynomials
Change the signs of the terms of the
polynomial being subtracted, and then
combine all the like terms.
ADDING AND SUBTRACTING POLYNOMIALS
Example
Add or subtract each of the following, as indicated.
1) (3x – 8) + (4x2 – 3x +3) = 3x – 8 + 4x2 – 3x + 3
= 4x2 + 3x – 3x – 8 + 3
= 4x2 – 5
2) 4 – (– y – 4) = 4 + y + 4 = y + 4 + 4 = y + 8
3) (– a2 + 1) – (a2 – 3) + (5a2 – 6a + 7)
= – a2 + 1 – a2 + 3 + 5a2 – 6a + 7
= – a2 – a2 + 5a2 – 6a + 1 + 3 + 7 = 3a2 – 6a + 11
ADDING AND SUBTRACTING POLYNOMIALS
In the previous examples, after discarding the
parentheses, we would rearrange the terms so
that like terms were next to each other in the
expression.
You can also use a vertical format in arranging
your problem, so that like terms are aligned with
each other vertically.
MULTIPLYING POLYNOMIALS
Let’s Multiply!
M U LT I P LY I N G P O LY N O M I A L S
Multiplying polynomials
• If all of the polynomials are monomials, use
the associative and commutative properties.
• If any of the polynomials are not monomials,
use the distributive property before the
associative and commutative properties.
Then combine like terms.
Multiplying Polynomials
Example
Multiply each of the following.
1) (3x2)(– 2x) = (3)(– 2)(x2 · x) = – 6x3
2) (4x2)(3x2 – 2x + 5)
= (4x2)(3x2) – (4x2)(2x) + (4x2)(5)
= 12x4 – 8x3 + 20x2
(Distributive property)
(Multiply the monomials)
3) (2x – 4)(7x + 5) = 2x(7x + 5) – 4(7x + 5)
= 14x2 + 10x – 28x – 20
= 14x2 – 18x – 20
Multiplying Polynomials
Example
Multiply (3x + 4)2
Remember that a2 = a · a, so (3x + 4)2 = (3x + 4)(3x + 4).
(3x + 4)2 = (3x + 4)(3x + 4) = (3x)(3x + 4) + 4(3x + 4)
= 9x2 + 12x + 12x + 16
= 9x2 + 24x + 16
Multiplying Polynomials
Example
Multiply (a + 2)(a3 – 3a2 + 7).
(a + 2)(a3 – 3a2 + 7) = a(a3 – 3a2 + 7) + 2(a3 – 3a2 + 7)
a4 – 3a3 + 7a + 2a3 – 6a2 +
14
= a4 – a3 – 6a2 + 7a + 14
=
Multiplying Polynomials
Example
Multiply (3x – 7y)(7x + 2y)
(3x – 7y)(7x + 2y) = (3x)(7x + 2y) – 7y(7x + 2y)
= 21x2 + 6xy – 49xy + 14y2
= 21x2 – 43xy + 14y2
Multiplying Polynomials
Example
Multiply (5x – 2z)2
(5x – 2z)2 = (5x – 2z)(5x – 2z) = (5x)(5x – 2z) – 2z(5x – 2z)
= 25x2 – 10xz – 10xz + 4z2
= 25x2 – 20xz + 4z2
Multiplying Polynomials
Example
Multiply (2x2 + x – 1)(x2 + 3x + 4)
(2x2 + x – 1)(x2 + 3x + 4)
= (2x2)(x2 + 3x + 4) + x(x2 + 3x + 4) – 1(x2 + 3x + 4)
=
2x4 + 6x3 + 8x2 + x3 + 3x2 + 4x – x2 – 3x – 4
=
2x4 + 7x3 + 10x2 + x – 4
SPECIAL PRODUCTS
Let’s multiply!
THE FOIL METHOD
When multiplying 2 binomials, the distributive
property can be easily remembered as the FOIL
method.
F – product of First terms
O – product of Outside terms
I – product of Inside terms
L – product of Last terms
Using the FOIL Method
Example
Multiply (y – 12)(y + 4)
(y – 12)(y + 4)
Product of First terms is y2
(y – 12)(y + 4)
Product of Outside terms is 4y
(y – 12)(y + 4)
Product of Inside terms is -12y
(y – 12)(y + 4)
Product of Last terms is -48
F O
I
L
(y – 12)(y + 4) = y2 + 4y – 12y – 48
= y2 – 8y – 48
Using the FOIL Method
Example
Multiply (2x – 4)(7x + 5)
L
F
F
O
I
L
(2x – 4)(7x + 5) = 2x(7x) + 2x(5) – 4(7x) – 4(5)
I
O
= 14x2 + 10x – 28x – 20
= 14x2 – 18x – 20
We multiplied these same two binomials together in the
previous section, using a different technique, but arrived at the
same product.
Special Products
In the process of using the FOIL method on products of
certain types of binomials, we see specific patterns that
lead to special products.
Squaring a Binomial
(a + b)2 = a2 + 2ab + b2
(a – b)2 = a2 – 2ab + b2
Multiplying the Sum and Difference of Two Terms
(a + b)(a – b) = a2 – b2
Special Products
Although you will arrive at the same
results for the special products by using
the techniques of this section or last
section, memorizing these products can
save you some time in multiplying
polynomials.
DIVIDING POLYNOMIALS
Let’s divide!
D I V I D I N G P O LY N O M I A L S
Dividing a polynomial by a monomial
Divide each term of the polynomial separately by the
monomial.
Example
12 a
3
36 a
3a
15
12 a
3
3a
4a
2
12
36 a
15
3a
3a
5
a
DIVIDING POLYNOMIALS
D I V I D I N G P O LY N O M I A L S
Dividing a polynomial by a polynomial
other than a monomial uses a “long
division” technique that is similar to the
process known as long division in
dividing two numbers, which is reviewed
on the next slide.
DIVIDING POLYNOMIALS
D I V I D I N G P O LY N O M I A L S
168
43 7256
43
29 5
258
37 6
344
32
Divide 43 into 72.
Multiply 1 times 43.
Subtract 43 from 72.
Bring down 5.
Divide 43 into 295.
Multiply 6 times 43.
Subtract 258 from 295.
Bring down 6.
Divide 43 into 376.
Multiply 8 times 43.
Subtract 344 from 376.
Nothing to bring down.
We then write our result as
168
32
.
43
Dividing
Polynomials
D I V I D I N G P O LY N O M I A L S
As you can see from the previous example, there is
a pattern in the long division technique.
Divide
Multiply
Subtract
Bring down
Then repeat these steps until you can’t bring
down or divide any longer.
We will incorporate this same repeated technique
with dividing polynomials.
DIVIDING POLYNOMIALS
D I V I D I N G P O LY N O M I A L S
7x
4x
5
3 28 x
2
23 x
15
28 x
2
12 x
Divide 7x into 28x2.
Multiply 4x times 7x+3.
Subtract 28x2 + 12x from 28x2 – 23x.
Bring down – 15.
35 x 15
Divide 7x into –35x.
Multiply – 5 times 7x+3.
Subtract –35x–15 from –35x–15.
35 x
Nothing to bring down.
15
So our answer is 4x – 5.
Dividing
Polynomials
D I V I D I N G P O LY N O M I A L S
2 x 10
2
2 x 7 4x 6x 8
2
4 x 14 x
20 x 8
20 x 70
78
We write our final answer as
Divide 2x into 4x2.
Multiply 2x times 2x+7.
Subtract 4x2 + 14x from 4x2 – 6x.
Bring down 8.
Divide 2x into –20x.
Multiply -10 times 2x+7.
Subtract –20x–70 from –20x+8.
Nothing to bring down.
2 x 10
78
( 2 x 7)
THE END
GOODBYE! 
Adding
and
Subtracting
6-4
6-4 Adding and Subtracting Polynomials
Polynomials
Warm Up
Lesson Presentation
Lesson Quiz
Holt
McDougal
Algebra
Holt
Algebra
1 1Algebra 1
6-4 Adding and Subtracting Polynomials
Warm Up
Simplify each expression by combining like
terms.
1. 4x + 2x
2. 8p – 5p
Simplify each expression.
3. 3(x + 4)
4. –1(x2 – 4x – 6)
Holt McDougal Algebra 1
6-4 Adding and Subtracting Polynomials
Essential Objective
Add and subtract polynomials.
Holt Algebra 1
6-4 Adding and Subtracting Polynomials
Just as you can perform operations on
numbers, you can perform operations on
polynomials.
To add or subtract polynomials, combine
like terms.
Holt Algebra 1
6-4 Adding and Subtracting Polynomials
Example: Adding and Subtracting Monomials
A. 12p3 + 11p2 + 8p3
12p3 + 11p2 + 8p3
20p3 + 11p2
B. 5x2 – 6 – 3x + 8
5x2 – 6 – 3x + 8
5x2 – 3x + 2
Holt Algebra 1
Identify like terms.
Combine like terms.
Identify like terms.
Combine like terms.
6-4 Adding and Subtracting Polynomials
I do….
C. t2 + 2s2 – 4t2 – s2
t2 + 2s2 – 4t2 – s2
t2 – 4t2 + 2s2 – s2
Identify like terms.
–3t2 + s2
D. 10m2n + 4m2n – 8m2n
10m2n + 4m2n – 8m2n
6m2n
Holt Algebra 1
Identify like terms.
6-4 Adding and Subtracting Polynomials
Remember!
Like terms are constants or terms with the same
variable(s) raised to the same power(s).
Holt Algebra 1
6-4 Adding and Subtracting Polynomials
We d0....
Add or subtract.
a. 2s2 + 3s2 + s
2s2 + 3s2 + s
5s2 + s
b. 4z4 – 8 + 16z4 + 2
4z4 – 8 + 16z4 + 2
4z4 + 16z4 – 8 + 2
20z4 – 6
Holt Algebra 1
6-4 Adding and Subtracting Polynomials
You do….
Add or subtract.
c. 2x8 + 7y8 – x8 – y8
x8 + 6y8
d. 9b3c2 + 5b3c2 – 13b3c2
b3c2
Holt Algebra 1
6-4 Adding and Subtracting Polynomials
Polynomials can be added in either vertical or
horizontal form.
In vertical form, align
the like terms and add:
5x2 + 4x + 1
+ 2x2 + 5x + 2
7x2 + 9x + 3
In horizontal form, use the
Associative and
Commutative Properties to
regroup and combine like
terms.
(5x2 + 4x + 1) + (2x2 + 5x + 2)
= (5x2 + 2x2) + (4x + 5x) + (1 + 2)
= 7x2 + 9x + 3
Holt Algebra 1
6-4 Adding and Subtracting Polynomials
Example: Adding Polynomials
Holt Algebra 1
6-4 Adding and Subtracting Polynomials
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 signs of each term:
Change the signs and proceed to addition!
–(2x3 – 3x + 7)= –2x3 + 3x – 7
Holt Algebra 1
6-4 Adding and Subtracting Polynomials
Example: Subtracting Polynomials
(x3 + 4y) – (2x3)
(x3 + 4y) + (–2x3)
Rewrite subtraction as addition
of the opposite.
(x3 + 4y) + (–2x3)
(x3 – 2x3) + 4y
–x3 + 4y
Holt Algebra 1
Combine like terms.
6-4 Adding and Subtracting Polynomials
We do….
Subtract.
(–10x2 – 3x + 7) – (x2 – 9)
(–10x2 – 3x + 7) + (–x2 + 9)
(–10x2 – 3x + 7) + (–x2 + 9)
–10x2 – 3x + 7
–x2 + 0x + 9
–11x2 – 3x + 16
Holt Algebra 1
6-4 Adding and Subtracting Polynomials
You do….
Subtract.
(9q2 – 3q) – (q2 – 5)
(9q2 – 3q) + (–q2 + 5)
(9q2 – 3q) + (–q2 + 5)
9q2 – 3q + 0
+ − q2 – 0q + 5
8q2 – 3q + 5
Holt Algebra 1
6-4 Adding and Subtracting Polynomials
Example: 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 + 11. Write a
polynomial that represents the total area of
both plots of land.
(3x2 + 7x – 5)
+ (5x2 – 4x + 11)
8x2 + 3x + 6
Holt Algebra 1
Plot A.
Plot B.
Combine like terms.
6-4 Adding and Subtracting Polynomials
Lesson Quiz
Add or subtract.
1. 7m2 + 3m + 4m2
2. (r2 + s2) – (5r2 + 4s2)
3. (10pq + 3p) + (2pq – 5p + 6pq)
4. (14d2 – 8) + (6d2 – 2d +1)
5. (2.5ab + 14b) – (–1.5ab + 4b)
Holt Algebra 1
01/06/17
SOUTHEAST MIDDLE SCHOOL
Horrel Hill Road, Hopkins, South Carolina
Name:__________________________Period:____________Date:______________
Math 8
Day 1 through 5
Lesson Plan: Recalling the Laws of Exponents
8.EEI.1 Understand and apply the laws of exponents (i.e., product
Standards
rule, quotient
rule, power to a power, product to a power, quotient to a power,
zero power
property, negative exponents) to simplify numerical expressions that
include
integer exponents.
Learning Targets I can apply the laws of exponents.
I Can Statements
Essential
Question(s)
How can the laws of exponents be applied in real-world situations?
Resources
Learning
Activities or
Experiences
1|P a g e
You will need a pair of scissors and a glue stick to complete this
assignment. All
answers should be written on the page provided.
1. Complete at least 3 topics of your ALEKS pathway. (if available)
Review attached notes and complete the “Exponent Rules
2. Puzzle.”
Practice
3. Assessment
by Dr. Antonio C. Ramirez, Jr.
Math 8 SY 2019-2020
Lesson Notes
(Please refer to the Powerpoint Presentation for details)
2|P a g e
by Dr. Antonio C. Ramirez, Jr.
Math 8 SY 2019-2020
Day 1 Activity Sheets
Challenge No. 1: Apply the laws of exponents in simplifying
these expressions. If you read the powerpoint presentation,
you will be able to get the correct answer.
Multiplication and Division Law
1. 3  3 
2
2
7.
s12

4
s
8.
39

5
3
2. 52  54 
3.
a a 
5
2
4. 2s 2  4s 7 
9.
s12 t 8

4 4
st
5. (3) 2  (3)3 
6.
3|P a g e
s 2t 4  s 7 t 3 
10.
by Dr. Antonio C. Ramirez, Jr.
36a 5b8

4 5
4a b
Math 8 SY 2019-2020
Day 2 Activity Sheets
Challenge No. 2 Apply the laws of exponents in simplifying
these expressions. If you read the powerpoint presentation,
you will get the correct answer.
Power to Power Law
 
1. 3
2 5
 
2. a
3 4
 
3. 2 a

2
5
s
7.   
t


2 3
3 
8.  5  
3 


5 3 2
4. 2 a b

5. (3a ) 
2 2
 
2 4 3
6. s t
4|P a g e
2
9

2
 st 
9.  4  
 rt 
8
2
 36a 5b8 
 
10. 
4 5 
 4a b 
by Dr. Antonio C. Ramirez, Jr.
Math 8 SY 2019-2020
Day 3 Activity Sheets
Challenge No. 3 Apply the laws of exponents in simplifying
these expressions. If you read the powerpoint presentation,
you will get the correct answer.
Negative and Zero Exponent Law
1.
2a b 
0
2
2.
2
y y
4
3.
a 

5 1



5. 3 x y
6.
5|P a g e

s t 
2 4 0



 

2
2
s t 
9.  4 4  
s t 
2 2
3 4
1
3 
8.  5  
3 
9
4. s 2  4s 7 
2
2
7. 
 x
2
2
 36a 
10.  4 5  
 4a b 
by Dr. Antonio C. Ramirez, Jr.
5
Math 8 SY 2019-2020
Day 4: Exponent Rules Puzzle
1. Cut out the nine puzzle pieces.
2. Pair up the matching expressions
(each non-simplified expression has a
matching simplified expression).
3. When complete, the puzzle will be a three-by-three square. Glue
your final arrangement on the page provided. GOOD LUCK!
6|P a g e
by Dr. Antonio C. Ramirez, Jr.
Math 8 SY 2019-2020
Page 3 of 5
7|P a g e
by Dr. Antonio C. Ramirez, Jr.
Math 8 SY 2019-2020
Exponent Rules Puzzle Solution
8|P a g e
by Dr. Antonio C. Ramirez, Jr.
Math 8 SY 2019-2020
Day 5 : Assessment
9|P a g e
by Dr. Antonio C. Ramirez, Jr.
Math 8 SY 2019-2020
10 | P a g e
by Dr. Antonio C. Ramirez, Jr.
Math 8 SY 2019-2020
Exponents
Power
5
exponent
3
base
Example: 125  53 means that 53 is the exponential
form of the number 125.
53 means 3 factors of 5 or 5 x 5 x 5
The Laws of Exponents:
#1: Exponential form: The exponent of a power indicates
how many times the base multiplies itself.
x  x  x  x x  x  x  x
n
n times
n factors of x
Example: 5  5  5  5
3
#2: Multiplying Powers:
If you are multiplying Powers
with the same base, KEEP the BASE & ADD the EXPONENTS!
x x  x
m
So, I get it!
When you
multiply
Powers, you
add the
exponents!
n
m n
2 6  23  2 6  3  29
 512
#3: Dividing Powers: When dividing Powers with the
same base, KEEP the BASE & SUBTRACT the EXPONENTS!
m
x
m
n
mn

x

x

x
n
x
So, I get it!
When you
divide
Powers, you
subtract the
exponents!
6
2
6 2
4

2

2
2
2
 16
Try these:
12
1. 3  3 
2
2
7.
2. 52  54 
3.
8.
a a 
5
2
4. 2s  4s 
2
7
12 8
9.
5. (3)  (3) 
2
6.
3
s t s t 
2 4
7 3
s

4
s
9
3

5
3
s t

4 4
st
5 8
10.
36a b

4 5
4a b
SOLUTIONS
2
2 2
4
a a  a
5 2
a
1. 3  3  3  3  81
2 4
6
2
4
2. 5  5  5  5
2
3.
5
2
4. 2s  4s  2  4  s
2
7
5. (3)  (3)  (3)
2
6.
3
s t s t 
2 4
7 3
s
7
2 7
23
 8s
 (3)  243
2 7 43
t
9
5
s t
9 7
SOLUTIONS
12
7.
8.
9.
10.
s
12  4
8
s

s

4
s
9
3
9 5
4
3

3

81

5
3
12 8
s t
12  4 8 4
8 4
s
t

s
t

4 4
st
5 8
36a b
5  4 8 5
3
36

4

a
b

9
ab

4 5
4a b
#4: Power of a Power: If you are raising a Power to an
exponent, you multiply the exponents!
x 
n
m
So, when I
take a Power
to a power, I
multiply the
exponents
x
x 
3 5
=
mn
x
35
=
15
x
#5: Product Law of Exponents: If the product of the
bases is powered by the same exponent, then the result is a
multiplication of individual factors of the product, each powered
by the given exponent.
 xy 
So, when I take
a Power of a
Product, I apply
the exponent to
all factors of
the product.
n
x y
n
n
(ab)  a b
2
2
2
#6: Quotient Law of Exponents: If the quotient of the
bases is powered by the same exponent, then the result is both
numerator and denominator , each powered by the given exponent.
n
 x
x
   n
y
 y
So, when I take a
Power of a
Quotient, I apply
the exponent to
all parts of the
quotient.
n
4
 2  2 16
   4 
81
3 3
4
Try these:
 
1. 3
5
2 5

 
3. 2a  
4. 2 a b 
2. a
s
7.   
t 2
 39 
8.  5  
3 
3 4
2 3
2
5 3 2
5. (3a ) 
2 2
 
2 4 3
6. s t


2
 st 
9.  4  
 rt 
5 8 2
 36a b 
 
10. 
4 5 
 4a b 
8
SOLUTIONS
 
1. 3
2 5
 
2. a
3 4

10
3

a12
 
 2 a


3. 2 a
2
2 3
3
5 3 2
4. 2 a b
23
 8a
6
 222 a 52b32  24 a10 b 6  16a10 b 6
5. (3a )  3  a 22  9a 4
2
2 2
 
2 4 3
6. s t
23 43
s t
s t
6 12
SOLUTIONS
5
s
7.   
t
5
s
5
t
2
3 
8.  5   34 2  38
3 
9
2
4 2
2 8
 st 


st
s
t
9.  4   
  2

r
 rt 
 r 
8
 36 a b
10 
4 5
 4a b
5 8
2

  9ab 3



2
2 32
9 a b
2
 81a b
2 6
#7: Negative Law of Exponents: If the base is powered
by the negative exponent, then the base becomes reciprocal with the
positive exponent.
So, when I have a
Negative Exponent, I
switch the base to its
reciprocal with a
Positive Exponent.
Ha Ha!
If the base with the
negative exponent is in
the denominator, it
moves to the
numerator to lose its
negative sign!
x
m
1
 m
x
1
1
5  3 
5
125
and
3
1
2

3
9
2
3
#8: Zero Law of Exponents: Any base powered by zero
exponent equals one.
x 1
0
So zero
factors of a
base equals 1.
That makes
sense! Every
power has a
coefficient
of 1.
50  1
and
a0  1
and
(5a ) 0  1
Try these:
1.
2a b 
0
2

2.
y 2  y 4 
3.
a 
5 1
2

4. s  4s 

7
 
s t  
2
5. 3 x y
6.
3 4
2 4 0
1
2 
7.   
 x 2
 39 
8.  5  
3 
2
2
s t 
9.  4 4  
s t 
2
5
 36a 
10.  4 5  
 4a b 
2 2
SOLUTIONS


0
1. 2 a b  1
2
1
3. a   5
a
5
2
7
4. s  4s  4s
5 1

2
5. 3 x y
 
2 4 0
6. s t

3 4

4
 3 x y
 1
8
12

8
x

81 y12
SOLUTIONS
1
2 
7.  
 x 
9 2
3 
8.  5 
3 
2
1
x
4
  
4
 x
 3
2

4 2
1
3  8
3
8
s t 
 2  2 2
4 4
9.  4 4   s t   s t
s t 
10
2
5
b

2

2
10
 36a 
9
a
b

2


10.  4 5  
81a
4
a
b


2 2