= 1 = -1

= x

= -x

= -x2

= x2

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Explore Modeling Binomial Multiplication Using algebra tiles to model the product of two binomials is very similar to using algebra tiles to model the product of a monomial and a polynomial.

Rules

1. The first factor goes on the left side of the grid, and the second factor goes on the top.

2. Fill in the grid with tiles that have the same height as tiles on the left and the same length as tiles on the top.

3. Follow the key. The product of two tiles of the same color is positive; the product of two tiles of different colors is negative.

Use algebra tiles to model (x + 1) (x - 2) . Then write the product. First fill in the factors and mat.

Now remove any zero pairs.

The product (x + 1) (x - 2) in simplest form is x 2 - x - .

Resource Locker

Module 5 177 Lesson 2

5.2 Multiplying Polynomial Expressions

Essential Question: How do you multiply binomials and polynomials?

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Evaluate: Homework and Practice

Multiply by using the Distributive Property.

1. (x + 6) (x - 4) 2. (2x + 5) (x - 3)

3. (x - 6) (x + 1) 4. ( x 2 + 3) (x - 4)

5. ( x 2 + 11) (x + 6) 6. ( x 2 + 8) (x - 5)

Multiply by using the FOIL method.

7. (x + 3) (x + 7) 8. (4x + 2) (x - 2)

9. (3x + 2) (2x + 5) 10. ( x 2 - 6) (x - 4)

11. ( x 2 + 9 ) (x - 3) 12. (4 x 2 - 4) (2x + 1)

Module 5 183 Lesson 2

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Multiply the polynomials.

13. (x - 3) ( x 2 + 2x + 1)

14. (x + 5) ( x 3 + 6 x 2 + 18x)

15. (x + 4) ( x 4 + x 2 + 1)

16. (x - 6) ( x 5 + 4 x 3 + 6 x 2 + 2x)

17. ( x 2 + x + 3) ( x 3 - x 2 + 4)

18. ( x 3 + x 2 + 2x) ( x 4 - x 3 + x 2 )

Module 5 184 Lesson 2

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Write a polynomial equation for each situation.

19. Gardening Cameron is creating a garden. He designs a rectangular garden with a length of (x + 6) feet and a width of (x + 2) feet. When x = 5, what is the area of the garden?

20. Design Sabrina has designed a rectangular painting that measures 50 feet in length and 40 feet in width. Alfred has also designed a rectangular painting, but it measures x feet shorter on each side. When x = 3, what is the area of Alfred’s painting?

21. Photography Karl is putting a frame around a rectangular photograph. The photograph is 12 inches long and 10 inches wide, and the frame is the same width all the way around. What will be the area of the framed photograph?

Module 5 185 Lesson 2

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22. Sports A tennis court is surrounded by a fence so that the distance from each boundary of the tennis court to the fence is the same. If the tennis court is 78 feet long and 36 feet wide, what is the area of the entire surface inside the fence?

23. State the first term of each product.

a. (2x + 1) (3x + 4) b. ( x 4 + x 2 ) (3 x 8 + x 11 ) c. x (x + 9) d. ( x 2 + 9) (3x + 4) (2x + 6) e. ( x 3 + 4) ( x 2 + 6) (x + 5)

24. Draw algebra tiles to model the factors in the polynomial multiplication modeled on the mat. Then write the factors and the product in simplest form.

Module 5 186 Lesson 2

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Explain 4 Modeling with Special Products

Example 4 Write and simplify an expression to represent the situation.

Design A designer adds a border with a uniform width to a square rug. The original side length of the rug is (x - 5) feet. The side length of the entire rug including the original rug and the border is (x + 5) feet. What is the area of the border? Evaluate the area of the border if x = 10 feet.

Analyze Information

Identify the important information.The answer will be an expression that represents the area of the border.List the important information:

The rug is a square with a side length of feet.The side length of the entire square area including the original rug and the border

is feet.

Formulate a Plan

The area of the rug in square feet is ( ) 2 . The total area of the rug plus the border in square feet

is ( ) 2 . The area of the rug can be subtracted from the total area to find the area of the border.

Solve

Find the total area:

(x + 5) 2 = x 2 + 2 ( ) ( ) + 2

= x 2 + x +

Find the area of the rug:

(x - 5) 2 = x 2 - 2 ( ) ( ) + ( ) 2 = x 2 - x +

Find the area of the border: Area of border = total area – area of rug

Area =  x 2 + x + - ( x 2 - x + ) =  x 2 + x + - x 2 + x -

= ( x 2 - x 2 ) + ( x + x) + ( - ) =  x 2 + x +

=  x

The area of the border is x 2 + x + = square feet.

Justify and Evaluate

Suppose that x = 10. The rug is feet by feet, so its area is square feet. The total area is

( + ) 2 = square feet, so the area of the border is - = square feet,

which is (10) when x = 10. So the answer makes sense.Module 5 192 Lesson 3

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Multiply.

1. (x + 8) 2 2. (4x + 6y) 2

3. (6 + x 2 ) 2 4. (-x + 5) 2

5. (x + 11) 2 6. (8x + 9y) 2

7. (x - 3) 2 8. (5x - 2) 2

9. (6x - 7y) 2 10. ( 5 - x 2 ) 2

11. (5x - 4y) 2 12. (7 - 2 x 2 ) 2

Evaluate: Homework and Practice

Module 5 194 Lesson 3

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13. (x + 4) (x - 4) 14. ( x 2 + 6y) ( x 2 - 6y)

15. (9 + x) (9 - x) 16. (2x + 5) (2x - 5)

17. ( 3x 2 + 8y) ( 3x 2 - 8y) 18. (7 + 3x) (7 - 3x)

Write and simplify an expression to represent the situation.

19. Design A square swimming pool is surrounded by a cement walkway with a uniform width. The swimming pool has a side length of (x - 2) feet. The side length of the entire square area including the pool and the walkway is (x + 1) feet. Write an expression for the area of the walkway. Then find the area of the cement walkway when x = 7 feet.

20. This week Leo worked (x + 4) hours at a pizzeria. He is paid (x - 4) dollars per hour. Leo’s friend Frankie worked the same number of hours, but he is paid (x - 2) dollars per hour. Write an expression for the total amount paid to the two workers. Then find the total amount if x = 12 dollars.

Module 5 195 Lesson 3

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21. Kyra is framing a square painting with side lengths of (x + 8) inches. The total area of the painting and the frame has a side length of (2x - 6) inches. The material for the frame will cost $0.08 per square inch. Write an expression for the area of the frame. Then find the cost of the material for the frame if x = 16.

22. Geometry Circle A has a radius of (x + 4) units. A larger circle, B, has a radius of (x + 5) units. Use the formula A = π r 2 to write an expression for the difference in the areas of the circles. Leave your answer in terms of π. Then use 3.14 for π to approximate to the nearest whole number the difference in the areas when x = 10.

23. A square has sides with lengths of (x - 1) units. A rectangle has a length of x units and a width of (x - 2) units. Which statements about the situation are true? Select all that apply.

a. The area of the square is ( x 2 - 1) square units.

b. The area of the rectangle is x 2 - 2x square units.

c. The area of the square is greater than the area of the rectangle.

d. The value of x must be greater than 2.

e. The difference in the areas is 2x - 1.

Module 5 196 Lesson 3

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