Transcript Algebra 1 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION Glencoe/McGraw-Hill 8787 Orion Place Columbus, Ohio 43240

Algebra 1 Interactive Chalkboard
Copyright © by The McGraw-Hill Companies, Inc.
Send all inquiries to:
GLENCOE DIVISION
Glencoe/McGraw-Hill
8787 Orion Place
Columbus, Ohio 43240
Lesson 8-1 Multiplying Monomials
Lesson 8-2 Dividing Monomials
Lesson 8-3 Scientific Notation
Lesson 8-4 Polynomials
Lesson 8-5 Adding and Subtracting Polynomials
Lesson 8-6 Multiplying Polynomials by a Monomial
Lesson 8-7 Multiplying Polynomials
Lesson 8-8 Special Products
Example 1 Identify Monomials
Example 2 Product of Powers
Example 3 Power of a Power
Example 4 Power of a Product
Example 5 Simplify Expressions
Determine whether each expression is a monomial.
Explain your reasoning.
Expression
Monomial?
a.
no
b.
yes
c.
yes
Reason
The expression involves
subtraction, not the product,
of two variables.
The expression is the
product of a number and
two variables.
is a real number and an
example of a constant.
d.
xy
yes
The expression is the product
of two variables.
Determine whether each expression is a monomial.
Explain your reasoning.
Expression
Monomial?
Reason
yes
Single variables are monomials.
b.
no
The expression involves
subtraction, not the product,
of two variables.
c.
no
The expression is the quotient,
not the product, of two variables.
a.
The expression is the product of a
d.
yes
number,
, and two variables.
Simplify
.
Commutative and
Associative Properties
Product of Powers
Answer:
Simplify.
Simplify
.
Communicative
and Associative
Properties
Product of Powers
Answer:
Simplify.
Simplify each expression.
a.
Answer:
b.
Answer:
Simplify
Power of a Power
Simplify.
Power of a Power
Answer:
Simplify.
Simplify
Answer:
Geometry Find the volume of a cube with a side
length
Volume
Formula for volume of a cube
Power of a Product
Answer:
Simplify.
Express the surface area of the
cube as a monomial.
Answer:
Simplify
Power of a Power
Power of a Product
Power of a Power
Commutative Property
Answer:
Power of Powers
Simplify
Answer:
Example 1 Quotient of Powers
Example 2 Power of a Quotient
Example 3 Zero Exponent
Example 4 Negative Exponents
Example 5 Apply Properties of Exponents
Simplify
Assume that x and y are not equal
to zero.
Group powers that have
the same base.
Quotient of Powers
Answer:
Simplify.
Simplify
to zero.
Answer:
Assume that a and b are not equal
Simplify
Assume that e and f are not
equal to zero.
Power of a Quotient
Power of a Product
Answer:
Power of a Power
Simplify
equal to zero.
Answer:
Assume that p and q are not
Simplify
equal to zero.
Answer: 1
Assume that m and n are not
Simplify
. Assume that m and n are not
equal to zero.
Simplify.
Answer:
Quotient of Powers
Simplify each expression. Assume that z is not equal
to zero.
a.
Answer: 1
b.
Answer:
Simplify
. Assume that y and z are not
equal to zero.
Write as a product
of fractions.
Answer:
Multiply fractions.
Simplify
. Assume that p, q, and r are
not equal to zero.
Group powers
with the same
base.
Quotient of
Powers and
Negative
Exponent
Properties
Simplify.
Negative
Exponent
Property
Answer:
Multiply
fractions.
Simplify each expression. Assume that no
denominator is equal to zero.
a.
Answer:
b.
Answer:
Multiple-Choice Test Item
Write the ratio of the circumference
of the circle to the area of the
square in simplest form.
A
B
C
D
Read the Test Item
A ratio is a comparison of two quantities. It can be written
in fraction form.
Solve the Test Item
•
•
circumference of a circle
length of a square diameter of circle or 2r
area of square
Substitute.
Quotient of
Powers
Simplify.
Answer: C
Multiple-Choice Test Item
Write the ratio of the circumference
of the circle to the perimeter of the
square in simplest form.
A
B
Answer: A
C
D
Example 1 Scientific to Standard Notation
Example 2 Standard to Scientific Notation
Example 3 Use Scientific Notation
Example 4 Multiplication with Scientific Notation
Example 5 Division with Scientific Notation
Express
in standard notation.
move decimal point 3
places to the left.
Answer: 0.00748
Express
in standard notation.
move decimal point 5
places to the right.
Answer: 219,000
Express each number in standard notation.
a.
Answer: 0.0316
b.
Answer: 7610
Express 0.000000672 in scientific notation.
Move decimal
point 7 places
to the right.
and
Answer:
Express 3,022,000,000,000 in scientific notation.
Move decimal point
12 places to the left.
and
Answer:
Express each number in scientific notation.
a. 458,000,000
Answer:
b. 0.0000452
Answer:
The Sporting Goods Manufacturers
Association reported that in 2000,
women spent $4.4 billion on 124
million pairs of shoes. Men spent $8.3
billion on 169 million pairs of shoes.
Express the numbers of pairs of shoes sold to
women, pairs sold to men, and total spent by both
men and women in standard notation.
Answer: Shoes sold to women:
Shoes sold to men:
Total spent:
Write each of these numbers in scientific notation.
Answer: Shoes sold to women:
Shoes sold to men:
Total spent:
The average circulation for all U.S. daily newspapers
in 2000 was 111.5 billion newspapers. The top three
leading newspapers were The Wall Street Journal,
with a circulation of 1.76 million newspapers, USA
Today, which sold 1.69 million newspapers, and The
New York Times, which had 1.10 million readers.
a. Express the average daily circulation and the
circulation of the top three newspapers in
standard notation.
Answer: Total circulation: 111,500,000,000; The Wall
Street Journal: 1,760,000; USA Today: 1,690,000; The
New York Times: 1,100,000
The average circulation for all U.S. daily newspapers
in 2000 was 111.5 billion newspapers. The top three
leading newspapers were The Wall Street Journal,
with a circulation of 1.76 million newspapers, USA
Today, which sold 1.69 million newspapers, and The
New York Times, which had 1.10 million readers.
b. Write each of the numbers in scientific notation.
Answer: Total circulation:
Journal:
USA Today:
The New York Times:
The Wall Street
Evaluate
Express the result in
scientific and standard notation.
Commutative and
Associative Properties
Product of Powers
Associative Property
Product of Powers
Answer:
Evaluate
Express the result in
scientific and standard notation.
Answer:
Evaluate
Express the result in scientific
and standard notation.
Associative Property
Product of Powers
Answer:
Evaluate
Express the result in scientific
and standard notation.
Answer:
Example 1 Identify Polynomials
Example 2 Write a Polynomial
Example 3 Degree of a Polynomial
Example 4 Arrange Polynomials in Ascending Order
Example 5 Arrange Polynomials in Descending Order
State whether each expression is a polynomial.
If it is a polynomial, identify it as a monomial,
binomial, or trinomial.
Expression
Polynomial?
Monomial,
Binomial, or
Trinomial
a.
Yes,
is the difference of two
real numbers.
binomial
b.
Yes,
is the sum and
difference of three monomials.
trinomial
c.
No.
d.
Yes,
are not monomials.
has one term.
none of these
monomial
State whether each expression is a polynomial.
If it is a polynomial, identify it as a monomial,
binomial, or trinomial.
Expression
Polynomial?
a.
Yes,
is the sum of
three monomials.
b.
No.
c.
Yes,
The
expression is the sum of two monomials.
d.
Yes,
which is not a monomial.
has one term.
Monomial,
Binomial, or
Trinomial
trinomial
none of these
binomial
monomial
Write a polynomial to
represent the area of the
green shaded region.
Words
The area of the shaded region is the area of
the rectangle minus the area of the triangle.
Variables area of the shaded region
height of rectangle
area of rectangle
triangle area
Equation
A
A
Answer: The polynomial representing the area of the
shaded region is
Write a polynomial to represent the
area of the green shaded region.
Answer:
Find the degree of each polynomial.
Degree of
Each Term
Degree of
Polynomial
a.
0, 1, 2, 3
3
b.
2, 1, 0
2
c.
8
8
Polynomial
Terms
Find the degree of each polynomial.
Degree of
Each Term
Degree of
Polynomial
a.
2 , 1, 3, 0
3
b.
2, 4 , 3
4
c.
7, 6
7
Polynomial
Terms
Arrange the terms of
powers of x are in ascending order.
Answer:
so that the
Arrange the terms of
that the powers of x are in ascending order.
Answer:
so
Arrange the terms of each polynomial so that the
powers of x are in ascending order.
a.
Answer:
b.
Answer:
Arrange the terms of
the powers of x are in descending order.
Answer:
so that
Arrange the terms of
so
that the powers of x are in descending order.
Answer:
Arrange the terms of each polynomial so that the
powers of x are in descending order.
a.
Answer:
b.
Answer:
Example 1 Add Polynomials
Example 2 Subtract Polynomials
Example 3 Subtract Polynomials
Find
Method 1 Horizontal
Group like terms together.
Associative and
Commutative
Properties
Add like terms.
Method 2 Vertical
Align the like terms in columns and add.
Notice that terms are in descending
order with like terms aligned.
Answer:
Find
Answer:
Find
Method 1 Horizontal
Subtract
by adding its additive inverse.
The additive inverse
of
is
Group like
terms.
Add like terms.
Method 2 Vertical
Align like terms in columns and subtract by adding the
additive inverse.
Add the opposite.
Answer:
or
Find
Answer:
Geometry The measure of
the perimeter of the triangle
shown is
Find the polynomial that represents the third side of
the triangle.
Let a = length of side 1, b = the length of side 2, and
c = the length of the third side.
You can find a polynomial for the third side by subtracting
side a and side b from the polynomial for the perimeter.
To subtract,
add the
additive
inverses.
Group the
like terms.
Add like
terms.
Answer: The polynomial for the third side is
Find the length of the third
side if the triangle if
The length of the third side is
Simplify.
Answer: 45 units
Geometry The measure of
the perimeter of the rectangle
shown is
a. Find a polynomial that represents width of
the rectangle.
Answer:
b. Find the width of the rectangle if
Answer: 3 units
Example 1 Multiply a Polynomial by a Monomial
Example 2 Simplify Expressions
Example 3 Use Polynomial Models
Example 4 Polynomials on Both Sides
Find
Method 1 Horizontal
Distributive Property
Multiply.
Find
Method 2 Vertical
Distributive Property
Multiply.
Answer:
Find
Answer:
Simplify
Distributive
Property
Product of
Powers
Commutative
and Associative
Properties
Combine like
terms.
Answer:
Simplify
Answer:
Entertainment Admission to the Super Fun
Amusement Park is $10. Once in the park, super rides
are an additional $3 each and regular rides are an
additional $2. Sarita goes to the park and rides 15
rides, of which s of those 15 are super rides.
Find an expression for how much money Sarita spent
at the park.
Words
The total cost is the sum of the admission,
super ride costs, and regular ride costs.
Variables If
the number of super rides, then
is the number of regular rides. Let M be the
amount of money Sarita spent at the park.
Equation
Amount
of money equals
M
super
admission plus rides
10
s
$3 per
regular
times ride
plus rides times
$2 per
ride.
3
2
Distributive Property
Simplify
Simplify.
Answer: An expression for the amount of money Sarita
spent in the park is
, where s is the number
of super rides she rode.
Evaluate the expression to find the cost if Sarita
rode 9 super rides.
Add.
Answer: Sarita spent $49.
The Fosters own a vacation home that they rent
throughout the year. The rental rate during peak
season is $120 per day and the rate during the off-peak
season is $70 per day. Last year they rented the house
210 days, p of which were during peak season.
a. Find an expression for how much rent the
Fosters received.
Answer:
b. Evaluate the expression if p is equal to 130.
Answer: $21,200
Solve
Original equation
Distributive Property
Combine like terms.
Subtract
from
each side.
Add 7 to each side.
Add 2b to each side.
Divide each side by 14.
Answer:
Check
Original
equation
Simplify.
Multiply.
Add and
subtract.
Solve
Answer:
Example 1 The Distributive Property
Example 2 FOIL Method
Example 3 FOIL Method
Example 4 The Distributive Property
Find
Method 1 Vertical
Multiply by –4.
Find
Multiply by y.
Find
Add like terms.
Find
Method 2 Horizontal
Distributive
Property
Distributive
Property
Multiply.
Combine like
terms.
Answer:
Find
Answer:
Find
F
L
F
O
I
L
I
O
Multiply.
Combine like
terms.
Answer:
Find
F
O
I
L
Multiply.
Answer:
Combine like
terms.
Find each product.
a.
Answer:
b.
Answer:
Geometry The area A of a
triangle is one-half the
height h times the base b.
Write an expression for the
area of the triangle.
Identify the height and the base.
Now write and apply the formula.
Area
A
equals
one-half
height
h
times
base.
b
Original formula
Substitution
FOIL method
Multiply.
Combine like terms.
Distributive Property
Answer: The area of the triangle is
square units.
Geometry The area of a
rectangle is the measure
of the base times the
height. Write an
expression for the area
of the rectangle.
Answer:
Find
Distributive
Property
Distributive
Property
Answer:
Combine
like terms.
Find
Distributive Property
Distributive Property
Answer:
Combine like terms.
Find each product.
a.
Answer:
b.
Answer:
Example 1 Square of a Sum
Example 2 Square of a Difference
Example 3 Apply the Sum of a Square
Example 4 Product of a Sum and a Difference
Find
Square of a Sum
Answer:
Simplify.
Check
Check your work by using the FOIL method.
F
O
I
L
Find
Square of a Sum
Answer:
Simplify.
Find each product.
a.
Answer:
b.
Answer:
Find
Square of a
Difference
Answer:
Simplify.
Find
Square of a
Difference
Answer:
Simplify.
Find each product.
a.
Answer:
b.
Answer:
Geometry Write an expression that represents
the area of a square that has a side length of
units.
The formula for the area of a square is
Area of a square
Simplify.
Answer: The area of the square is
square units.
Geometry Write an expression that represents
the area of a square that has a side length of
units.
Answer:
Find
Product of a Sum
and a Difference
Answer:
Simplify.
Find
Product of a Sum
and a Difference
Answer:
Simplify.
Find each product.
a.
Answer:
b.
Answer:
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