18.2 Multiplying Polynomial Expressions

Transcription

1 Name Class Date 18. Multiplying Polynomial Expressions Essential Question: How do you multiply binomials and polynomials? Resource Locker 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.. 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 - ). Then write the product. First fill in the factors and mat. Key = x = -x Now remove any zero pairs. = x = -x = 1 = -1 The product (x + 1) (x - ) in simplest form is x - x -. Module Lesson

2 Reflect 1. Discussion Why can zero pairs be removed from the product?. Discussion Is it possible for more than one pair of tiles to form a zero pair? Explain 1 Multiplying Binomials Using the Distributive Property To multiply a binomial by a binomial, the Distributive Property must be applied more than once. Example 1 Multiply by using the Distributive Property. A (x + 5) (x + ) (x + 5) (x + ) = x (x + ) + 5 (x + ) Distribute. = x (x + ) + 5 (x + ) Redistribute and simplify. = x (x) + x () + 5 (x) + 5 () = x + x + 5x + 10 = x + 7x + 10 B (x + 4) (x + 3) (x + 4) (x + 3) = x (x + 3) + (x + 3) Distribute. Your Turn 3. (x + 1) (x - ) = x (x + 3) + (x + 3) Redistribute and simplify. = x (x) + (3) + (x) + (3) = x + x + x + = x + x + Module Lesson

3 Explain Multiplying Binomials Using FOIL Another way to use the Distributive Property is the FOIL method. The FOIL method uses the Distributive Property to multiply terms of binomials in this order: First terms, Outer terms, Inner terms, and Last terms. Example Multiply by using the FOIL method. A ( x + 3) (x + ) Use the FOIL method. (x + 3) (x + ) = ( x + 3) (x + ) F Multiply the first terms. Result: x 3 = ( x + 3) (x + ) O Multiply the outer terms. Result: x = ( x + 3) (x + ) I Multiply the inner terms. Result: 3x = ( x + 3) (x + ) L Multiply the last terms. Result: 6 Add the result. ( x + 3) (x + ) = x 3 + x + 3x + 6 B (3 x - x) (x + 5) Use the FOIL method. ( 3x - x) (x + 5) = ( 3x - x) (x + 5) F Multiply the first terms. Result: = ( 3x - x) (x + 5) O Multiply the outer terms. Result: = ( 3x - x) (x + 5) I Multiply the inner terms. Result: = ( 3x - x) (x + 5) L Multiply the last terms. Result: Add the result. (3x - x) (x + 5) = x 3 + x - x Reflect 4. The FOIL method finds the sum of four partial products. Why does the result from part B only have three terms? 5. Can the FOIL method be used for numeric expressions? Give an example. Your Turn 6. (x + 3) (x + 6) Module Lesson

4 Explain 3 Multiplying Polynomials To multiply polynomials with more than two terms, the Distributive Property must be used several times. Example 3 Multiply the polynomials. A (x + ) ( x - 5x + 4) (x + ) ( x - 5x + 4) = x (x - 5x + 4) + ( x - 5x + 4) Distribute. = x ( x - 5x + 4) + ( x - 5x + 4) Redistribute. = x (x ) + x (-5x) + x (4) + ( x ) + (-5x) + (4) Simplify. = x 3-5 x + 4x + x - 10x + 8 = x 3-3 x - 6x + 8 B (3x - 4) (- x + 5x - 6) (3x - 4) (- x + 5x - 6) = 3x (- x + 5x - 6) - (- x + 5x - 6) Distribute. = 3x (- x + 5x - 6) - (- x + 5x - 6) Redistribute. Simplify. = 3x (- x ) + 3x ( ) + 3x ( ) - 4 ( ) - 4 ( ) - 4 = x + x - x + x - x + = x + x - x + Reflect 7. Discussion Is the product of two polynomials always another polynomial? 8. Can the Distributive Property be used to multiply two trinomials? 9. (3x + 1) ( x x - 7) Module Lesson

5 Explain 4 Modeling with Polynomial Multiplication Polynomial multiplication is sometimes necessary in problem solving. A Gardening Trina is building a garden. She designs a rectangular garden with length (x + 4) feet and width (x + 1) feet. When x = 4, what is the area of the garden? Let y represent the area of Trina s garden. Then the equation for this situation is y = (x + 4) (x + 1). y = (x + 4) (x + 1) Use FOIL. y = x + x + 4x + 4 y = x + 5x + 4 Now substitute 4 for x to finish the problem. y = x + 5x + 4 y = (4) + 5 (4) + 4 y = y = 40 The area of Trina s garden is 40 ft. Image Credits: Tim Pannell/Corbis B Design Orik has designed a rectangular mural that measures 0 feet in width and 30 feet in length. Laura has also designed a rectangular mural, but it measures x feet shorter on each side. When x = 6, what is the area of Laura s mural? Let y represent the area of Laura s mural. Then the equation for this situation is y = (0 - x) (30 - x). y = (0 - x) (30 - x) Use FOIL. y = - x - x + x y = x - x + Now substitute y = - + for x to finish the problem. y = - + y = The area of Laura s mural is ft. Module Lesson

6 Your Turn 10. Landscaping A landscape architect is designing a rectangular garden in a local park. The garden will be 0 feet long and 15 feet wide. The architect wants to place a walkway with a uniform width all the way around the garden. What will be the area of the garden, including the walkway? v Elaborate 11. How is the FOIL method different from the Distributive Property? Explain. 1. Why can FOIL not be used for polynomials with three or more terms? 13. Essential Question Check In How do you multiply two binomials? Module Lesson

7 Evaluate: Homework and Practice Multiply by using the Distributive Property. 1. (x + 6) (x - 4). (x + 5) (x - 3) Online Homework Hints and Help Extra Practice 3. (x - 6) (x + 1) 4. (x + 3) (x - 4) 5. (x + 11) (x + 6) 6. (x + 8) (x - 5) Multiply by using the FOIL method. 7. (x + 3) (x + 7) 8. (4x + ) (x - ) 9. (3x + ) (x + 5) 10. (x - 6) (x - 4) 11. (x + 9 ) (x - 3) 1. (4 x - 4) (x + 1) Module Lesson

8 Multiply the polynomials. 13. (x - 3) ( x + x + 1) 14. (x + 5) ( x x + 18x) 15. (x + 4) ( x 4 + x + 1) 16. (x - 6) ( x x x + x) 17. ( x + x + 3) ( x 3 - x + 4) 18. ( x 3 + x + x) ( x 4 - x 3 + x ) Module Lesson

9 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 + ) feet. When x = 5, what is the area of the garden? 0. 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? Image Credits: Paul Burns/Corbis 1. Photography Karl is putting a frame around a rectangular photograph. The photograph is 1 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 Lesson

10 . 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? 3. State the first term of each product. a. (x + 1) (3x + 4) b. ( x 4 + x )(3 x 8 + x 11 ) c. x (x + 9) d. ( x + 9) (3x + 4) (x + 6) e. ( x 3 + 4) ( x + 6) (x + 5) 4. 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. Image Credits: Berna Namoglu/Shutterstock Module Lesson

11 H.O.T. Focus on Higher Order Thinking 5. Critical Thinking The product of 3 consecutive odd numbers is 145. Write an expression for finding the numbers. 6. Represent Real-World Problems The town swimming pool is d feet deep. The width of the pool is 10 feet greater than 5 times its depth. The length of the pool is 35 feet greater than 5 times its depth. Write and simplify an expression to represent the volume of the pool. 7. Explain the Error Bill argues that (x + 1) (x + 19) simplifies to x + 0x + 0. Explain his error. Module Lesson

12 Lesson Performance Task Roan is planning a large vegetable garden in her yard. She plans to have at least six x by x regions for rotating crops and some or 3 feet by x strips for fruit bushes like blueberries and raspberries. Design a rectangular garden for Roan and write a polynomial that will give its area. Module Lesson