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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 Contents

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 Lesson 1 Contents

Determine whether each expression is a monomial. Explain your reasoning. xy d. c. b. a. Reason Monomial? Expression no The expression involves subtraction, not the product, of two variables. yes The expression is the product of a number and two variables. yes is a real number and an example of a constant. yes The expression is the product of two variables. Example 1-1a

Determine whether each expression is a monomial. Explain your reasoning. b. a. Reason Monomial? Expression yes Single variables are monomials. no The expression involves subtraction, not the product, of two variables. no The expression is the quotient, not the product, of two variables. yes The expression is the product of a number, , and two variables. Example 1-1b

Commutative and Associative Properties Simplify . Commutative and Associative Properties Product of Powers Simplify. Answer: Example 1-2a

Commutative and Associative Properties Simplify . Commutative and Associative Properties Product of Powers Simplify. Answer: Example 1-2b

Simplify each expression. a. b. Answer: Answer: Example 1-2c

Simplify Power of a Power Simplify. Power of a Power Simplify. Answer: Example 1-3a

Simplify Answer: Example 1-3b

Geometry Find the volume of a cube with a side length Formula for volume of a cube Power of a Product Simplify. Answer: Example 1-4a

Express the surface area of the cube as a monomial. Answer: Example 1-4b

Simplify Power of a Power Power of a Product Power of a Power Example 1-5a

Commutative Property Answer: Power of Powers Example 1-5b

Simplify Answer: Example 1-5c

End of Lesson 1

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 Lesson 2 Contents

Simplify Assume that x and y are not equal to zero. Group powers that have the same base. Quotient of Powers Answer: Simplify. Example 2-1a

Simplify Assume that a and b are not equal to zero. Answer: Example 2-1b

Simplify Assume that e and f are not equal to zero. Power of a Quotient Power of a Product Power of a Power Answer: Example 2-2a

Simplify Assume that p and q are not equal to zero. Answer: Example 2-2b

Simplify Assume that m and n are not equal to zero. Answer: 1 Example 2-3a

Simplify . Assume that m and n are not equal to zero. Answer: Quotient of Powers Example 2-3b

Simplify each expression. Assume that z is not equal to zero. b. Answer: 1 Answer: Example 2-3c

Simplify . Assume that x, y, and z are not equal to zero. Write as a product of fractions. Answer: Multiply fractions. Example 2-4a

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 Example 2-4b

Negative Exponent Property Simplify. Negative Exponent Property Multiply fractions. Answer: Example 2-4c

Simplify each expression. Assume that no denominator is equal to zero. b. Answer: Answer: Example 2-4d

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. Example 2-5a

Solve the Test Item circumference of a circle side length of the square diameter of circle or 2r area of square Substitute. Quotient of Powers Example 2-5b

Simplify. Answer: C Example 2-5c

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 Answer: A Example 2-5d

End of Lesson 2

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 Lesson 3 Contents

Express in standard notation. move decimal point 3 places to the left. Answer: 0.00748 Example 3-1a

Express in standard notation. move decimal point 5 places to the right. Answer: 219,000 Example 3-1b

Express each number in standard notation. a. Answer: 0.0316 Answer: 7610 Example 3-1c

Express 0.000000672 in scientific notation. Move decimal point 7 places to the right. and Answer: Example 3-2a

Express 3,022,000,000,000 in scientific notation. Move decimal point 12 places to the left. and Answer: Example 3-2b

Express each number in scientific notation. a. 458,000,000 Answer: Answer: Example 3-2c

Answer: Shoes sold to women: 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: Example 3-3a

Write each of these numbers in scientific notation. Answer: Shoes sold to women: Shoes sold to men: Total spent: Example 3-3b

The average circulation for all U. S. daily newspapers in 2000 was 111 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 Example 3-3c

b. Write each of the numbers in scientific notation. 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: 1.76 USA Today: The New York Times: Example 3-3d

Evaluate Express the result in scientific and standard notation. Commutative and Associative Properties Product of Powers Associative Property Example 3-4a

Product of Powers Answer: Example 3-4b

Evaluate Express the result in scientific and standard notation. Answer: Example 3-4c

Evaluate Express the result in scientific and standard notation. Associative Property Product of Powers Answer: Example 3-5a

Evaluate Express the result in scientific and standard notation. Answer: Example 3-5b

End of Lesson 3

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 Lesson 4 Contents

Monomial, Binomial, or Trinomial State whether each expression is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial. Monomial, Binomial, or Trinomial Polynomial? Expression a. b. c. d. Yes, is the difference of two real numbers. binomial Yes, is the sum and difference of three monomials. trinomial No. are not monomials. none of these Yes, has one term. monomial Example 4-1a

Monomial, Binomial, or Trinomial State whether each expression is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial. Monomial, Binomial, or Trinomial Polynomial? Expression a. b. c. d. Yes, is the sum of three monomials. trinomial No. which is not a monomial. none of these Yes, The expression is the sum of two monomials. binomial Yes, has one term. monomial Example 4-1b

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 Example 4-2a

Equation A A Answer: The polynomial representing the area of the shaded region is Example 4-2b

Write a polynomial to represent the area of the green shaded region. Answer: Example 4-2c

Find the degree of each polynomial. b. a. Degree of Polynomial Degree of Each Term Terms Polynomial 0, 1, 2, 3 3 2, 1, 0 2 8 8 Example 4-3a

Find the degree of each polynomial. b. a. Degree of Polynomial Degree of Each Term Terms Polynomial 2ac2, –7 2, 1, 3, 0 3 2, 4, 3 4 – 7, 6 7 – Example 4-3b

Arrange the terms of so that the powers of x are in ascending order. Answer: Example 4-4a

Arrange the terms of so that the powers of x are in ascending order. Answer: Example 4-4b

Arrange the terms of each polynomial so that the powers of x are in ascending order. b. Answer: 1 – 2x + 6x2 – 3x4 Answer: Example 4-4c

Arrange the terms of so that the powers of x are in descending order. Answer: Example 4-5a

Arrange the terms of so that the powers of x are in descending order. Answer: Example 4-5b

Arrange the terms of each polynomial so that the powers of x are in descending order. b. Answer: Answer: Example 4-5c

End of Lesson 4

Example 1 Add Polynomials Example 2 Subtract Polynomials Lesson 5 Contents

Group like terms together. Find Method 1 Horizontal Group like terms together. Associative and Commutative Properties Add like terms. Example 5-1a

Align the like terms in columns and add. Method 2 Vertical Align the like terms in columns and add. Notice that terms are in descending order with like terms aligned. Answer: Example 5-1b

Find Answer: Example 5-1c

Subtract by adding its additive inverse. Find Method 1 Horizontal Subtract by adding its additive inverse. The additive inverse of is Group like terms. Add like terms. Example 5-2a

Method 2 Vertical Align like terms in columns and subtract by adding the additive inverse. Add the opposite. Answer: or Example 5-2b

Find Answer: Example 5-2c

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. Example 5-3a

To subtract, add the additive inverses. Example 5-3b

Answer: The polynomial for the third side is Group the like terms. Add like terms. Answer: The polynomial for the third side is Example 5-3c

Find the length of the third side if the triangle if The length of the third side is Simplify. Answer: 45 units Example 5-3d

a. Find a polynomial that represents the width of the rectangle. Geometry The measure of the perimeter of the rectangle shown is 10r – 60. a. Find a polynomial that represents the width of the rectangle. b. Find the width of the rectangle if Answer: Answer: 3 units Example 5-3e

End of Lesson 5

Example 1 Multiply a Polynomial by a Monomial Example 2 Simplify Expressions Example 3 Use Polynomial Models Example 4 Polynomials on Both Sides Lesson 6 Contents

Distributive Property Find Method 1 Horizontal Distributive Property Multiply. Example 6-1a

Distributive Property Find Method 2 Vertical Distributive Property Multiply. Answer: Example 6-1b

Find Answer: Example 6-1c

Distributive Property Simplify Distributive Property Product of Powers Commutative and Associative Properties Combine like terms. Answer: Example 6-2a

Simplify Answer: Example 6-2b

Find an expression for how much money Sarita spent at the park. 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. Example 6-3a

M 10 s 3 2 Equation Distributive Property Simplify. Simplify. Amount of money equals admission plus super rides times $3 per ride regular rides $2 per ride. M 10 s 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. Example 6-3b

Evaluate the expression to find the cost if Sarita rode 9 super rides. Add. Answer: Sarita spent $49. Example 6-3c

a. Find an expression for how much rent the Fosters received. 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: Answer: $21,200 Example 6-3d

Distributive Property Solve Original equation Distributive Property Combine like terms. Subtract from each side. Example 6-4a

Add 7 to each side. Add 2b to each side. Divide each side by 14. Answer: Example 6-4b

Check Original equation Simplify. Multiply. Add and subtract. Example 6-4c

Solve Answer: Example 6-4d

End of Lesson 6

Example 1 The Distributive Property Example 2 FOIL Method Lesson 7 Contents

Find Method 1 Vertical Multiply by –4. Example 7-1a

Find Multiply by y. Example 7-1b

Find Add like terms. Example 7-1c

Distributive Property Find Method 2 Horizontal Distributive Property Distributive Property Multiply. Combine like terms. Answer: Example 7-1d

Find Answer: Example 7-1e

Find F L O I Multiply. Combine like terms. Answer: Example 7-2a

Find F I O L Multiply. Answer: Combine like terms. Example 7-2b

Find each product. a. b. Answer: Answer: Example 7-2c

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 Example 7-3a

Original formula Substitution FOIL method Multiply. Example 7-3b

Distributive Property Combine like terms. Distributive Property Answer: The area of the triangle is square units. Example 7-3c

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: Example 7-3d

Distributive Property Find Distributive Property Distributive Property Answer: Combine like terms. Example 7-4a

Distributive Property Find Distributive Property Distributive Property Answer: Combine like terms. Example 7-4b

Find each product. a. b. Answer: Answer: Example 7-4c

End of Lesson 7

Example 2 Square of a Difference Example 3 Apply the Sum of a Square 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 Lesson 8 Contents

Find Square of a Sum Answer: Simplify. Example 8-1a

Check Check your work by using the FOIL method. Example 8-1b

Find Square of a Sum Answer: Simplify. Example 8-1c

Find each product. a. b. Answer: Answer: Example 8-1d

Find Square of a Difference Answer: Simplify. Example 8-2a

Find Square of a Difference Answer: Simplify. Example 8-2b

Find each product. a. b. Answer: Answer: Example 8-2c

The formula for the area of a square is 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. Example 8-3a

Geometry Write an expression that represents the area of a square that has a side length of units. Answer: Example 8-3b

Product of a Sum and a Difference Find Product of a Sum and a Difference Answer: Simplify. Example 8-4a

Product of a Sum and a Difference Find Product of a Sum and a Difference Answer: Simplify. Example 8-4b

Find each product. a. b. Answer: Answer: Example 8-4c

End of Lesson 8

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