Algebra 1
Chapter 8
Mr. Zaluckyj
Name: ____________________
1
Algebra 1
Chapter 8
Learning Targets
Learning Targets
1
2
3
1
2
3
1
2
Got it!
Goal Sheet
Ouch!
Need more
practice
Got it!
Correct on
Test
I can condense/expand an
expression using exponents.
I can simplify expressions using
the product of power property.
(same place powers)
I can simplify expressions using
the power of power property.
(super powers)
I can simplify expressions
involving the product of quotient
property. (different place
powers)
I can simplify expressions
containing negative exponents.
(elevator power)
I can simplify expressions
containing an exponent of zero.
(zero power)
I can express numbers in
scientific notation and standard
notation.
I can find products and
quotients of numbers expressed
in scientific notation.
1
I can find the degree of a
polynomial.
2
I can arrange the terms of a
polynomial in descending order.
1
I can add polynomials.
2
I can subtract polynomials.
1
I can find the product of a
monomial and a polynomial.
2
I can solve equations involving
polynomials.
1
I can multiply two polynomials by
using.
2
8.1 Notes
Laws of Exponents: Multiplying Monomials
Warm up
Write each expression as a power.
A) (9  9  9)
__________________
Expand each expression.
A) x 4 = __________
B) (5)(5)(5)(5) __________________
B) 6 2 = ____________
C) x  y  y  y  y  y __________________
C) (5)2 = ___________
D) ( xy )( xy )( xy )( xy ) __________________
2
D)   = _____________
5
x3
E) 2 = ____________
y
1 1 1
E) 5   
4 4 4
3
___________________
The exponent is 4
The exponent is 4
2   2  2  2  2
4
The base is +2
(2)4  (2)  (2)  (2)  (2)
The base is -2
Investigate: Try to find the pattern for multiplying powers…..
Expand
Simplify
1) x  x
2
2) x  x
2
2
3) x  x
3
4) x  x
4
2
2
What about x  x
3
100
?
“Same Place Rule”:
Product – of - Power Property:
xm  xn 
3
Ex. 1 Rewrite each.
a.
c.
a4  a3 
y 2y

x 2x 4
b.
x 3y 4y 2

2 6
ax x a
d.
aba 3

5 4 2 2
abba
Investigate: Try to find the pattern for raising a power to a power…..
Expand
1)
x 
2)
x 
3)
x 
Simplify
2 3
4 2
3 3
 
What about x
3 100
?
“SUPER POWER Rule”:
 xm  
n
Power – of – a - Power Property:
Ex. 2 Simplify
a.
x 
4 5
b.
 2x 
3 2
c.
 3ab 
2 3
d. (2 x y )(3xy )
3
4
3 2
Mixed:
d.
(2 x 2 )(4 x3 y 2 )
e. (3a b)(6ab c)
2
4
f.
(11c8 )(10c 4 d )
HW: ______________________
4
8.1 Homework
Simplify
1. (4a4b)(9a2b3)
2. (ab4 )(ab2 )
3. (3 j 7 k 5 )(8 jk 8 )
4. (5a2b3c4)(6a3b4c2)
5. (7c3d 4 )(4cd 3 )
6. (5x2)(2xy2)(-2y4)
7. (x2)4
8. (9pq7)2
9. (4cd)2(3d3)
10. (2x5)3(-5xy6)
11. (3m2n3)3(m3n)4
5
8.2 Notes
Dividing Monomials
Warm up: Choose the best answer to represent each rule then state the name of the
rule.
____1. Simplify
a)
x m n
b)
____2. Simplify
a)
x m x n ? Name of the rule _________________________
c)
xmn
d)
x m n
d)
x m n
(x m )n ? Name of the rule _________________________
x m n
3. Simplify
x m n
b)
x m n
c)
xmn
2xy   3y x  =
4
2
3
Investigate: Try to find the pattern with dividing monomials
Expand
1)
x3
x2
2)
x6
x3
3)
x10
x3
4)
x150
x 20
Simplify
“Different Place Rule”:
Product – of - Quotient Property:
xm

n
x
6
Example 1:
x5
1. 2
x
Simplify
x5 y 3
2. 4
x y
 2a 4b 
5. 

 a 
4a 7 b 7 c 2
4.
8b3a3c 2
14 y 7
3.
2 y5
3
 4 p 4 q5 
6.  3 2 
 3p q 
2
Investigate:
Expand
1)
x4
x5
2)
x2
x6
Simplify
“Elevator Rule”:
Negative Exponent Property:
x m 
Overview:
Anything raised to a negative power on top ________________________________________
Anything raised to a negative power on bottom ____________________________________
Anything raised to a power of 0 is __________.
Example 2: Simplify.
x2
7. 3
x
p 8
8.
p3
b 4
9. 5
b
x y
1
10.
4w1 y 2
0
11.
 3xb  u 4
x 1b2u 7
Homework: __________________________
7
8.2 HOMEWORK
1. (4ac2)(-5a4c2)
2. ( -3d2f 6)(-4d3f4)
3. a-3t5(at-9)
18 j 4 k 7
4.
9 j 3k 2
3m 4 n 7
5.
12m 7 n 6
6. 4p-3(5p-2)
24u 3v 4 w1
7.
8u 2v 3 w2
8.  4 x3 y
 2a b 
 4a b 
3 5 3
10.
3 2 2
  2x y 
2 2
5
11.  c 4 d 5 f 5   c 4 d 2 f 6 
3
 2mn 
4m n  n m 
2 3
2
9.
12.
6
4
2
3 0
( x 4 y 3 )2 (a99 )0
x9 y 6
8
8.1-8.2 REVIEW
State the 5 EXPONENT Rules on the line and then use the rule to simplify each.
Name of Rule
Example of the Rule
1.
a 
2.
a0 
3.
2 2 
4.
x5

x3
5.
x2 x3 
2
Simplify each expression.
1. a 2 a 3
2. ( x 3 )3
6. ( y z )
2 3 2
6x5z
11.
12x9 z 3
7. (cd ) (c d )
t 3s9
12.
(ts)5
2
 15c 6 
8. 

 3c 
4
13. (2df )(2d f )
3
2
4
2

5. (2r 2 s) 4
4. (5r 3 )(2r 2 s)
3. (4a 2 )(3a3 )
3
5
 d 2 z6 
9. 
3 
 5dz 
2
 11d 100 f 

14. 
10 78 
 21d f 
0
10.
15.
6t 8 s 6
18t 5 s 2
(2 x 2 ) 3 (10 x 4 )
8x
QUIZ tomorrow 
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8.3 Scientific Notation
Warm up:
Simplify each
1.
12x 4 y 3z
20x 8y 5z 2
3. 3
2
2. (2x 4y 3 )2 (7x 4y )(8x 7 y 6 )
4.
(97x 81y 2 )0
2x 
3
8.3.1 I can express numbers in scientific notation and standard notation.
8.3.2 I can find products and quotients of numbers expressed in scientific notation.
When you deal with very large numbers like 5,000,000 or very small numbers like 0.00000005, it is
difficult to keep track of how many zeroes there are and it’s also tough to tell if one number is bigger
or smaller than another number when they’re written normally. So we write a number in scientific
notation, which is when a number is written as
a x 10 n
Scientific Notation
Standard Notation
Ex 1) Express each number in standard form.
A. 6.32 x 105
Standard Notation
where 1  a  10 and n is a whole number.
B. 7.8 x 10-6
Scientific Notation
Ex 2) Express each number in scientific notation.
A. 5,120,000
B.
.000475
10
To multiply or divide numbers in scientific notation, you follow the same type of rules we did to
multiply monomials with numbers and variables.
So just like we did before, you’re going to multiply or divide the like terms.
2x4 · 5x3 =
Ex 4) Simplify each scientific notation expression.
A) (2.3 x 103)·(4 x 106) =
B)
13  108

4  103
C) 37.6  104 
Daily life in the United States
Ex 5) For the problems below, consider the population of the United States 250 million people
and consider a year to have 365 days.
12
A. Approximately 1.095 X 10
gallons of sewage is dumped off the coasts of the United States
yearly.
-How much is dumped each day?
-How much is dumped each hour?
-How much is dumped each minute?
B. Approximately 7.7 X 10
given per person?
10
dollars is given to charities each year. What is the average amount
HOMEWORK: ________________________
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8.3 Homework
Write each answer in scientific notation.
1. (3 x 104)(2 x 10-2)
4.
2. (2 x 109)(2.1 x 102)
2.4  10 4
1.2  10 2
5.
12 103
2 108
3. (1.3 x 10-2)(4.2 x 105)
6. (4.8 x 10-5)÷(4 x 10-1)
Write in scientific notation.
7. 124,000,000
8. 50,000,000,000
9. .00000567
10. The weight of the Earth is 6,600,000,000,000,000,000,000 tons. Write the weight in scientific
notation.
11. The size of a cell is 2.5 X 10
12. Approximately 1.6 x 10
thrown away each day?
10
3
centimeters in diameter, how small is that?
disposable diapers are thrown into the trash each year. How many are
12
8.4-8.6 Notes
Adding/Subtracting Polynomials & Multiplying a polynomial by a monomial
Warm up
4 103
1) Simplify: A.
2.3 105
B)
x 2y 8
x 3y 9

C) 2 x5 y 3
3xy 
2 3
Polynomial: An expression of more than two algebraic terms. Monomial, Binomial, Trinomial…
Ex 1) Simplify each. Write final answer in descending order and then determine the degree of the
polynomial.
A) 6 x 2 (4 x  2)  2 x 2
B)
Degree of Polynomial ____
Degree of Polynomial ____
C)
D)
Degree of the Polynomial __________
E)
 4  3a    7a
3
3
 a  11
Degree of the Polynomial __________
7x
2
 6 x  10    3  5 x  4 x 2 
*
Degree of the Polynomial _________
F)
6x
4
 4    2 x 4  10 
Degree of the Polynomial _________
Homework:________________________
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8.4-8.6 HOMEWORK
Simplify these expressions! Make sure to write your answer in descending order.
1)
2)
 5 y 2  3 y  8   4 y 2  9 
 6a 2  7a  9   2a  5a 2  a  10 
Degree of the Polynomial __________
Degree of the Polynomial __________
12x  y    4x  5 y    y  6x 
3)
Degree of the Polynomial __________
Degree of the Polynomial __________
(3  a 4  2a )  (a 4  8a  5)
5)

 
Degree of the Polynomial __________
(3y 2  2)  (5  7y  3y 2 )
6)
Degree of the Polynomial __________
7) 3 7 y 3  y  10  2 4y 3  4y 2  7
2 g  g 3  2 g 2  5g  6    g 2  2 g 
4)
Degree of the Polynomial __________




8) 3y 2y 3  4 y  6  2y 8y 3  4 y 2

Degree of the Polynomial __________
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8.4-8.6 Continued… Notes
Warm Up
Simplify.
1)
2)
3x( 2 x 2  5)  x(3x  4  5x 2 )
Essential Question: How is this similar or different from what we learned before?
Learning Targets: Students will be able to… Find the product of a monomial and a polynomial.
Solve equations using polynomials.
Multiply and then simplify if necessary.
Ex 1.
Solve each equation.
Ex 3.
Ex 2.
Ex 4.
You Try:
Ex 5.
15
Ex 6. The Grayslake Park is the shape of a square. If they want to make an extension to the park 5
feet in each direction, write an expression that represents the new perimeter.
Ex 7)Find the area of the rectangle.
3x 2
3x  4
Homework: _____________________
16
Homework 8.4-8.6 continued…
Multiply and then simplify if necessary.
1. -4x(8+3x)
2. 5y(-2y 2 -7y)
3. 7ag(g 3 +2ag)
4. -3np(n 2 -2p)
6. –x(4x 2 -2x)-5x 3
5. d(-2d+4) + 15d
7. 3w(6w-4) + 2(w 2 -3w+5)
8. 5n(2n 3 +n 2 +8) +n(4-n)
9)  8 y 2  y  10   2  2 x 2  6 y  5 
10) 2  7 x 2  x  5    9 x 2  x  1
Solve each equation.
11. 2(4x-7) = 5(-2x-9) - 5
12. 2(5a-12) = -6(2a-3) + 2
Simplify.
13. 10(4m 3 -3m+2) - 2m(-3m 2 -7m+1)
Solve the equation.
14. 3g(g-4) - 2g(g-7) = g(g+6) - 28
17
8.7-8.8 Notes Multiplying Polynomials
Simplify the following!
1.
 6a c  a b c 
14 3
3 3
Warm up:
2.
4
3. 5x 2y (3xy  4xy 2 )
Example: Distribute:
 16a 2b 3 

4 8 
 8a b 
3
4. 6y 2 (4y 2  y  1)  7y (2y  3)
(x  3)(x  2) .
(x  3)(x  2)
Ex 1: Find each product.
A. (x  3)(x  8)
D.
 x  4
2
B. (3x  4)(7x  5)
E)
 6 p  1
C.
3x  7 y 3x  7 y 
2
18
Ex 2) Simplify the following.
A.
 x  2 (3x2  5x  4)
B. (2x  3)(6x 2  7x  1)
C. Find the area of the triangle. The area formula for a triangle is A 
1
bh .
2
x+2
6x 2 + 2x+ 8
HOMEWORK: _______________
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8.7-8.8 HOMEWORK
Simplify by distributing.
1. (x  6)(x  9)
2. (3r  3)(2r  1)
4.
 4 y 15 y  6 
7.
3m  3m  6   3  m2  4m  1
5.  3 x  2 
3. (5x  4)(2x  8)
2
6.
8.
9  4x 9  4 x 
(2 x 2  3)(2 x 2  3x  4)
9) The Grayslake Park is the shape of a square. If they want to make an extension to the park 5 feet
in each direction, write an expression that represents the new area.
20