text 8. polynomials
DESCRIPTION
Notes by Dr. David Archerteacher of Calculus at Andress HighTRANSCRIPT
ALGEBRA PROJECT
UNIT 8
POLYNOMIALS
POLYNOMIALS
Lesson 1 Multiplying Monomials
Lesson 2 Dividing Monomials
Lesson 3 Scientific Notation
Lesson 4 Polynomials
Lesson 5 Adding and Subtracting Polynomials
Lesson 6 Multiplying Polynomials by a Monomial
Lesson 7 Multiplying Polynomials
Lesson 8 Special Products
MULTIPLY MONOMIALS
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
xyd.
c.
b.
a.
ReasonMonomial?Expression
Determine whether each expression is a monomial. Explain your reasoning.
The expression is the product of two variables.
yes
yes
The expression is the product of a number and two variables.
yes
The expression involves subtraction, not the product, of two variables.
no
is a real number and an
example of a constant.
d.
c.
b.
a.
ReasonMonomial?Expression
Determine whether each expression is a monomial. Explain your reasoning.
yes
no
The expression involves subtraction, not the product, of two variables.
no
Single variables are monomials.yes
The expression is the quotient, not the product, of two variables.
The expression is the product of a
number, , and two variables.
Simplify .
Commutative and Associative Properties
Product of Powers
Simplify.Answer:
Simplify .
Communicative and Associative Properties
Product of Powers
Simplify.Answer:
Simplify each expression.
a.
b.
Answer:
Answer:
Simplify
Simplify.Answer:
Power of a Power
Simplify.
Power of a Power
Simplify
Answer:
Geometry Find the volume of a cube with a side length
Simplify.Answer:
Volume Formula for volume of a cube
Power of a Product
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:
DIVIDING MONOMIALS
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.
Quotient of Powers
Group powers that have the same base.
Answer: Simplify.
Simplify Assume that a and b are not equal
to zero.
Answer:
Simplify Assume that e and f are not
equal to zero.
Power of a Quotient
Power of a Product
Power of a PowerAnswer:
Simplify Assume that p and q are not
equal to zero.
Answer:
Simplify Assume that m and n are not
equal to zero.
Answer: 1
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.
b.
Answer: 1
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.
Multiply fractions.
Answer:
Negative Exponent Property
Simplify each expression. Assume that no denominator is equal to zero.
a.
b.
Answer:
Answer:
Read the Test Item A ratio is a comparison of two quantities. It can be written in fraction form.
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
Solve the Test Item
• circumference of a circlelength of a square diameter of circle or 2rarea of square
• Substitute.
Quotient of Powers
Simplify.
Answer: C
Answer: A
Multiple-Choice Test Item
Write the ratio of the circumference of the circle to the perimeter of the square in simplest form.
A B C D
SCIENTIFIC NOTATION
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
Answer: 219,000
Express in standard notation.
move decimal point 5 places to the right.
Express each number in standard notation.
a.
b.
Answer: 0.0316
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.
Answer:
and
Express each number in scientific notation.
a. 458,000,000
b. 0.0000452
Answer:
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 thecirculation 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: The Wall Street Journal: USA Today: The New York Times:
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:
POLYNOMIALS
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
Monomial, Binomial, or
TrinomialPolynomial?Expression
a.
b.c.
d.
Yes, has one term.
State whether each expression is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial.
monomial
none of these
trinomial
binomialYes, is the difference of two real numbers.
Yes, is the sum and difference of three monomials.
No. are not monomials.
Monomial, Binomial, or
TrinomialPolynomial?Expression
a.
b.c.
d.
State whether each expression is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial.
monomial
binomial
none of these
trinomialYes, is the sum ofthree monomials.
No. which is not a monomial.
Yes, has one term.
Yes, The expression is the sum of two monomials.
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 regionheight 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:
c.
b.
a.
Degree of Polynomial
Degree of Each Term
TermsPolynomial
Find the degree of each polynomial.
88
22, 1, 0
30, 1, 2, 3
c.
b.
a.
Degree of Polynomial
Degree of Each Term
TermsPolynomial
Find the degree of each polynomial.
77, 6
42, 4, 3
32, 1, 3, 0
Arrange the terms of so that the powers of x are in ascending order.
Answer:
Arrange the terms of so that the powers of x are in ascending order.
Answer:
Arrange the terms of each polynomial so that the powers of x are in ascending order.
a.
b.
Answer:
Answer:
Arrange the terms of so that the powers of x are in descending order.
Answer:
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.
b.
Answer:
Answer:
ADDING AND SUBTRACTING POLYNOMIALS
Example 1 Add Polynomials
Example 2 Subtract Polynomials
Example 3 Subtract Polynomials
Find
Method 1 HorizontalGroup like terms together.
Associative and Commutative Properties
Add like terms.
Method 2 Vertical
Notice that terms are in descending order with like terms aligned.
Answer:
Align the like terms in columns and add.
Find
Answer:
Method 1 Horizontal
Find
Subtract by adding its additive inverse.
The additive inverseofis
Group like terms.
Add like terms.
Method 2 VerticalAlign 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 ofthe 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.
Add like terms.
Answer: The polynomial for the third side is
Group the like terms.
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.
b. Find the width of the rectangle if
Answer:
Answer: 3 units
MULTIPLYING POLYNOMIALS by a MONOMIAL
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.
Method 2 Vertical
Distributive Property
Multiply.
Find
Answer:
Find
Answer:
Simplify
Distributive Property
Product of Powers
Commutative and Associative Properties
Answer:
Combine like terms.
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, thenis the number of regular rides. Let M be the amount of money Sarita spent at the park.
Equation
Amountof money equals admission plus
superrides times
$3 perride plus
regularrides times
$2 perride.
M 10 s 3 2
Answer: An expression for the amount of money Saritaspent in the park is , where s is the number of super rides she rode.
Distributive Property
Simplify
Simplify.
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.
b. Evaluate the expression if p is equal to 130.
Answer: $21,200
Answer:
Solve
Subtract from each side.
Original equation
Distributive Property
Combine like terms.
Add 7 to each side.
Add 2b to each side.
Divide each side by 14.
Answer:
Original equation
Check
Add and subtract.
Simplify.
Multiply.
Solve
Answer:
MULTIPLY POLYNOMIALS
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
Answer:
Distributive Property
Multiply.
Combine like terms.
Distributive Property
Find
Answer:
Find
Multiply.
Combine like terms.
Answer:
F F
O
O
I
IL L
Find
Multiply.
Answer: Combine like terms.
F IO L
Find each product.
a.
b.
Answer:
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 equals one-half height times base.
A h 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 arectangle is the measure of the base times theheight. Write anexpression for the areaof the rectangle.
Answer:
Find
Distributive Property
Distributive Property
Answer: Combine like terms.
Answer:Combine like terms.
Find
Distributive Property
Distributive Property
Find each product.
a.
b.
Answer:
Answer:
SPECIAL PRODUCTS
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
Square of a Sum
Find
Answer: Simplify.
Find each product.
a.
b.
Answer:
Answer:
Find
Square of a Difference
Answer: Simplify.
Square of a Difference
Answer: Simplify.
Find
Find each product.
a.
b.
Answer:
Answer:
Geometry Write an expression that representsthe 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 representsthe 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.
b.
Answer:
Answer:
THIS IS THE ENDOF THE SESSION
BYE!