text 8. polynomials

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!